Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #511583 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C2×C42 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #32707 |
| commutator, Frattini, cosocle | C22×C8 | C24 |
| prime radical | #585 | D4 |
| LCS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C22×C4 ◁ #3573 ◁ G | |
| LECS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
585 conjugacy classes of subgroups; only the 110 normal subgroups are shown.
This group has 585 conjugacy classes of subgroups — too many to draw. Only its 110 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 128 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 4 generators on 128 points:
(1,2,8,16,9,14,6,13)(3,17,22,10,23,47,20,45)(4,24,29,50,30,11,27,52)(5,12,33,55,35,56,36,58)(7,38,41,59,42,15,37,62)(18,65,69,43,70,63,68,102)(19,53,31,44,73,104,74,105)(21,60,39,100,77,64,75,46)(25,81,76,78,84,108,83,48)(26,79,32,91,87,49,88,54)(28,51,40,61,66,80,89,99)(34,94,93,117,97,57,98,115)(67,119,71,101,121,103,122,111)(72,118,90,127,95,112,124,106)(82,125,85,113,123,109,126,107)(86,110,92,114,96,116,120,128);(1,3,9,23)(2,10,14,45)(4,25,30,84)(5,31,35,74)(6,22,8,20)(7,39,42,75)(11,48,24,78)(12,53,56,104)(13,47,16,17)(15,60,38,64)(18,66,70,28)(19,36,73,33)(21,37,77,41)(26,85,87,126)(27,76,29,83)(32,82,88,123)(34,95,97,72)(40,68,89,69)(43,80,102,51)(44,55,105,58)(46,59,100,62)(49,109,79,125)(50,81,52,108)(54,113,91,107)(57,106,94,127)(61,65,99,63)(67,120,121,92)(71,96,122,86)(90,98,124,93)(101,114,111,128)(103,110,119,116)(112,117,118,115);(1,4,9,30)(2,11,14,24)(3,18,23,70)(5,32,35,88)(6,29,8,27)(7,40,42,89)(10,43,45,102)(12,54,56,91)(13,52,16,50)(15,61,38,99)(17,63,47,65)(19,71,73,122)(20,69,22,68)(21,76,77,83)(25,75,84,39)(26,36,87,33)(28,37,66,41)(31,67,74,121)(34,96,97,86)(44,103,105,119)(46,81,100,108)(48,64,78,60)(49,58,79,55)(51,59,80,62)(53,111,104,101)(57,116,94,110)(72,123,95,82)(85,124,126,90)(92,98,120,93)(106,113,127,107)(109,118,125,112)(114,117,128,115);(1,5)(2,12)(3,19)(4,26)(6,36)(7,34)(8,33)(9,35)(10,44)(11,49)(13,58)(14,56)(15,57)(16,55)(17,53)(18,67)(20,74)(21,72)(22,31)(23,73)(24,79)(25,82)(27,88)(28,86)(29,32)(30,87)(37,98)(38,94)(39,90)(40,92)(41,93)(42,97)(43,101)(45,105)(46,106)(47,104)(48,107)(50,91)(51,110)(52,54)(59,117)(60,118)(61,114)(62,115)(63,103)(64,112)(65,119)(66,96)(68,122)(69,71)(70,121)(75,124)(76,85)(77,95)(78,113)(80,116)(81,125)(83,126)(84,123)(89,120)(99,128)(100,127)(102,111)(108,109)Polycyclic presentation:
2^9|a^2=h,b^2=i,c^2=i,e^2=i,g^2=hi,h^2=i,b^a=be,c^a=cf,f^a=fi,g^a=gi,c^b=cg,d^b=def,e^b=ei,g^b=gh,h^b=hi,d^c=de,e^c=ei,g^c=gh,h^c=hi,e^d=ei44 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 418781);