Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6792404 |
| 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 | |
1797 conjugacy classes of subgroups.
| minimal degree | μ(G) = 64 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 5 generators on 64 points:
(1,2,7,12)(3,8,15,29)(4,9,19,33)(5,10,23,37)(6,11,26,40)(13,41,43,27)(14,28,44,55)(16,30,24,38)(17,46,45,31)(18,32,48,57)(20,34,25,39)(21,35,51,60)(22,36,53,62)(42,63,64,54)(47,59,50,56)(49,58,52,61);(1,3)(2,8)(4,17)(5,21)(6,24)(7,15)(9,31)(10,35)(11,38)(12,29)(13,20)(14,22)(16,26)(18,42)(19,45)(23,51)(25,43)(27,34)(28,36)(30,40)(32,54)(33,46)(37,60)(39,41)(44,53)(47,52)(48,64)(49,50)(55,62)(56,61)(57,63)(58,59);(1,4)(2,9)(3,13)(5,18)(6,25)(7,19)(8,27)(10,32)(11,39)(12,33)(14,42)(15,43)(16,45)(17,24)(20,26)(21,50)(22,52)(23,48)(28,54)(29,41)(30,46)(31,38)(34,40)(35,59)(36,61)(37,57)(44,64)(47,51)(49,53)(55,63)(56,60)(58,62);(1,3)(2,8)(4,13)(5,14)(6,16)(7,15)(9,27)(10,28)(11,30)(12,29)(17,20)(18,42)(19,43)(21,22)(23,44)(24,26)(25,45)(31,34)(32,54)(33,41)(35,36)(37,55)(38,40)(39,46)(47,49)(48,64)(50,52)(51,53)(56,58)(57,63)(59,61)(60,62);(1,5,6,22,7,23,26,53)(2,10,11,36,12,37,40,62)(3,14,16,21,15,44,24,51)(4,18,25,52,19,48,20,49)(8,28,30,35,29,55,38,60)(9,32,39,61,33,57,34,58)(13,42,45,50,43,64,17,47)(27,54,46,59,41,63,31,56)Polycyclic presentation:
2^9|a^2=i,e^2=fh,h^2=i,b^a=bf,c^a=cg,d^a=df,c^b=ch,e^b=efh,g^b=gi,h^b=hi,f^c=fi,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, 6699603);