Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6792410 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C28 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #38904 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C22×C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1776 conjugacy classes of subgroups.
| minimal degree | μ(G) = 64 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 5 generators on 64 points:
(1,2)(3,9)(4,19)(5,11)(6,29)(7,12)(8,13)(10,23)(14,34)(15,31)(16,40)(17,32)(18,33)(20,48)(21,47)(22,41)(24,36)(25,37)(26,54)(27,38)(28,39)(30,44)(35,51)(42,58)(43,57)(45,61)(46,56)(49,64)(50,62)(52,59)(53,55)(60,63);(1,3,8,18)(2,9,13,33)(4,20,22,44)(5,25,28,45)(6,17,7,16)(10,34,36,57)(11,37,39,61)(12,40,29,32)(14,23,43,24)(15,26,46,27)(19,30,41,48)(21,42,50,63)(31,54,56,38)(35,55,59,64)(47,60,62,58)(49,52,53,51);(1,4)(2,10)(3,14)(5,21)(6,24)(7,23)(8,22)(9,30)(11,35)(12,41)(13,36)(15,42)(16,20)(17,44)(18,43)(19,29)(25,53)(26,52)(27,51)(28,50)(31,55)(32,57)(33,48)(34,40)(37,60)(38,62)(39,59)(45,49)(46,63)(47,54)(56,64)(58,61);(1,2)(3,9)(4,10)(5,11)(6,29)(7,12)(8,13)(14,30)(15,31)(16,40)(17,32)(18,33)(19,24)(20,34)(21,35)(22,36)(23,41)(25,37)(26,54)(27,38)(28,39)(42,55)(43,48)(44,57)(45,61)(46,56)(47,52)(49,58)(50,59)(51,62)(53,60)(63,64);(1,5,6,26,8,28,7,27)(2,11,29,54,13,39,12,38)(3,15,16,45,18,46,17,25)(4,21,24,52,22,50,23,51)(9,31,40,61,33,56,32,37)(10,35,19,47,36,59,41,62)(14,42,20,49,43,63,44,53)(30,55,34,58,48,64,57,60)Polycyclic presentation:
2^9|b^2=i,e^2=fhi,g^2=i,h^2=i,b^a=bf,c^a=cg,d^a=df,g^a=gi,c^b=ch,e^b=efh,g^b=gi,h^b=hi,f^c=fi,g^c=gi,h^c=hi,g^d=gi62 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6699609);