Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #7624184 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C28 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C23×D8 |
| commutator, Frattini, cosocle | C2×C8 | C25 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ #525 ◁ #2929 ◁ G | |
| LECS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
1492 conjugacy classes of subgroups.
| minimal degree | μ(G) = 72 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 72 points:
(1,2)(3,7)(4,6)(5,8)(9,10)(11,16)(12,25)(13,29)(14,34)(15,18)(17,26)(19,32)(20,31)(21,45)(22,37)(23,51)(24,54)(27,60)(28,57)(30,44)(33,56)(35,43)(36,47)(38,53)(39,50)(40,59)(41,61)(42,58)(46,68)(48,71)(49,67)(52,66)(55,65)(62,69)(63,72)(64,70);(1,3)(2,5)(4,8)(6,7)(9,11,15,22)(10,16,18,37)(12,21,28,46)(13,23,33,49)(14,24,31,50)(17,36,40,62)(19,38,42,64)(20,39,34,54)(25,45,57,68)(26,47,59,69)(27,48,35,55)(29,51,56,67)(30,52,61,72)(32,53,58,70)(41,63,44,66)(43,65,60,71);(9,12,15,28)(10,17,18,40)(11,21,22,46)(13,30,33,61)(14,35,31,27)(16,36,37,62)(19,43,42,60)(20,44,34,41)(23,52,49,72)(24,55,50,48)(25,56,57,29)(26,58,59,32)(38,65,64,71)(39,66,54,63)(45,67,68,51)(47,70,69,53);(3,8)(5,7)(10,18)(11,22)(17,40)(19,42)(20,34)(21,46)(23,49)(24,50)(25,57)(29,56)(41,44)(43,60)(47,69)(48,55)(52,72)(53,70);(2,6)(5,7)(11,22)(16,37)(21,46)(23,49)(24,50)(36,62)(38,64)(39,54)(45,68)(47,69)(48,55)(51,67)(52,72)(53,70)(63,66)(65,71)Polycyclic presentation:
2^9|b^2=i,c^2=i,g^2=hi,h^2=i,b^a=bf,c^a=cg,d^a=di,e^a=ef,g^a=gh,h^a=hi,d^b=dfi,e^b=ei,g^c=gh,h^c=hi65 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7531383);