Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #506267 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C25 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #28143 |
| commutator, Frattini, cosocle | C22×C8 | C24 |
| prime radical | #575 | D4 |
| LCS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C22×C4 ◁ #2955 ◁ G | |
| LECS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
534 conjugacy classes of subgroups; only the 128 normal subgroups are shown.
This group has 534 conjugacy classes of subgroups — too many to draw. Only its 128 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)(3,16)(4,10)(5,27)(6,14)(7,34)(8,15)(9,19)(11,28)(12,33)(13,36)(17,54)(18,60)(20,67)(21,40)(22,70)(23,74)(24,46)(25,78)(26,47)(29,50)(30,87)(31,44)(32,51)(35,92)(37,64)(38,62)(39,71)(41,66)(42,68)(43,75)(45,80)(48,84)(49,104)(52,90)(53,79)(55,61)(56,69)(57,94)(58,96)(59,95)(63,97)(65,91)(72,120)(73,99)(76,81)(77,105)(82,86)(83,102)(85,93)(88,124)(89,101)(98,113)(100,121)(103,127)(106,108)(107,119)(109,110)(111,115)(112,128)(114,117)(116,118)(122,123)(125,126);(1,3,7,20,33,66,79,71,8,21,36,69,6,19,32,65)(2,9,13,39,12,16,51,56,15,41,34,91,14,40,53,67)(4,22,25,64,31,89,90,59,26,73,80,58,24,68,35,17)(5,18,30,63,84,113,104,119,29,62,86,115,28,61,85,114)(10,42,45,95,44,70,92,96,47,101,78,54,46,99,52,37)(11,38,49,97,27,55,82,107,48,60,93,111,50,98,87,117)(23,72,77,112,83,126,127,109,76,121,123,108,75,116,88,57)(43,100,103,128,74,118,122,110,102,120,124,106,81,125,105,94);(1,4,8,26)(2,10,15,47)(3,17,21,59)(5,23,29,76)(6,31,33,24)(7,35,36,90)(9,37,41,96)(11,43,48,102)(12,46,14,44)(13,52,34,92)(16,54,40,95)(18,57,62,109)(19,64,66,58)(20,68,69,89)(22,71,73,65)(25,79,80,32)(27,81,50,74)(28,83,84,75)(30,88,86,127)(38,94,60,110)(39,99,91,70)(42,56,101,67)(45,51,78,53)(49,105,93,122)(55,106,98,128)(61,112,113,108)(63,116,115,126)(72,119,121,114)(77,104,123,85)(82,124,87,103)(97,125,111,118)(100,107,120,117);(1,5,8,29)(2,11)(3,18,21,62)(4,23,26,76)(6,28,33,84)(7,30,36,86)(9,38)(10,43)(12,27)(13,49)(14,50)(15,48)(16,55)(17,57,59,109)(19,61,66,113)(20,63,69,115)(22,72,73,121)(24,75,31,83)(25,77,80,123)(32,85,79,104)(34,93)(35,88,90,127)(37,94)(39,97)(40,98)(41,60)(42,100)(44,74)(45,103)(46,81)(47,102)(51,82)(52,105)(53,87)(54,106)(56,107)(58,108,64,112)(65,114,71,119)(67,117)(68,116,89,126)(70,118)(78,124)(91,111)(92,122)(95,128)(96,110)(99,125)(101,120)Polycyclic presentation:
2^9|b^2=g,c^2=i,d^2=eh,e^2=i,g^2=hi,h^2=i,b^a=be,c^a=cf,d^a=de,e^a=ei,g^a=gi,c^b=cg,e^c=ei,g^c=gh,h^c=hi,f^d=fi62 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 413465);