Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #506543 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C23×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #28341 |
| commutator, Frattini, cosocle | C22×C8 | C24 |
| prime radical | #3584 | C22 |
| LCS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C22×C4 ◁ #1757 ◁ G | |
| LECS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
698 conjugacy classes of subgroups; only the 140 normal subgroups are shown.
This group has 698 conjugacy classes of subgroups — too many to draw. Only its 140 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:
(1,2,9,38,36,13,35,42,8,16,7,14,6,15,18,57)(3,17,39,95,75,47,74,45,23,46,22,62,21,10,41,94)(4,24,63,50,27,89,37,11,30,79,28,59,29,58,40,51)(5,12,33,54,91,52,64,109,34,56,32,53,31,55,90,97)(19,65,82,99,67,48,76,60,69,113,25,43,68,106,81,77)(20,61,72,116,123,104,96,101,73,103,71,115,70,44,100,102)(26,83,86,107,84,111,92,49,88,78,85,112,87,110,93,108)(66,118,80,127,119,128,124,114,122,117,120,98,121,105,125,126);(1,3,18,41,6,21,7,22,8,23,35,74,36,75,9,39)(2,10,42,95,15,46,38,94,16,47,57,62,13,17,14,45)(4,25,37,19,29,76,63,68,30,82,40,69,27,81,28,67)(5,20,64,96,31,70,33,72,34,73,90,100,91,123,32,71)(11,48,89,99,50,65,24,77,51,106,58,43,59,113,79,60)(12,44,97,115,55,103,53,101,56,104,109,116,52,61,54,102)(26,80,93,66,87,125,85,121,88,120,92,122,84,124,86,119)(49,105,110,127,107,117,78,126,108,128,111,98,112,118,83,114);(1,4)(2,11)(3,19)(5,26)(6,27)(7,37)(8,30)(9,40)(10,43)(12,49)(13,50)(14,58)(15,59)(16,51)(17,60)(18,63)(20,66)(21,67)(22,76)(23,69)(24,42)(25,74)(28,35)(29,36)(31,84)(32,92)(33,93)(34,88)(38,89)(39,81)(41,82)(44,98)(45,65)(46,77)(47,99)(48,95)(52,107)(53,110)(54,111)(55,112)(56,108)(57,79)(61,114)(62,106)(64,85)(68,75)(70,119)(71,124)(72,125)(73,122)(78,97)(80,100)(83,109)(86,90)(87,91)(94,113)(96,120)(101,118)(102,117)(103,126)(104,127)(105,115)(116,128)(121,123);(1,5)(2,12)(3,20)(4,26)(6,31)(7,33)(8,34)(9,32)(10,44)(11,49)(13,52)(14,54)(15,55)(16,56)(17,61)(18,64)(19,66)(21,70)(22,72)(23,73)(24,78)(25,80)(27,84)(28,86)(29,87)(30,88)(35,90)(36,91)(37,93)(38,53)(39,71)(40,92)(41,96)(42,97)(43,98)(45,102)(46,103)(47,104)(48,105)(50,107)(51,108)(57,109)(58,111)(59,112)(60,114)(62,116)(63,85)(65,117)(67,119)(68,121)(69,122)(74,100)(75,123)(76,125)(77,126)(79,83)(81,124)(82,120)(89,110)(94,101)(95,115)(99,127)(106,128)(113,118)Polycyclic presentation:
2^9|a^2=fi,b^2=egi,e^2=i,f^2=hi,g^2=hi,h^2=i,b^a=be,c^a=cf,d^a=deh,e^a=ei,c^b=cg,f^c=fh,g^c=gh,h^c=hi,f^d=fi50 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 413742);