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 | |
1800 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)(5,10,24,38)(6,11,22,36)(13,41,43,27)(14,28,44,55)(16,30,25,39)(17,46,45,31)(18,32,48,57)(20,34,26,40)(21,35,51,60)(23,37,52,61)(42,63,64,54)(47,59,50,56)(49,58,53,62);(1,3)(2,8)(4,17)(5,21)(6,25)(7,15)(9,31)(10,35)(11,39)(12,29)(13,20)(14,23)(16,22)(18,42)(19,45)(24,51)(26,43)(27,34)(28,37)(30,36)(32,54)(33,46)(38,60)(40,41)(44,52)(47,53)(48,64)(49,50)(55,61)(56,62)(57,63)(58,59);(1,4)(2,9)(3,13)(5,18)(6,26)(7,19)(8,27)(10,32)(11,40)(12,33)(14,42)(15,43)(16,45)(17,25)(20,22)(21,50)(23,53)(24,48)(28,54)(29,41)(30,46)(31,39)(34,36)(35,59)(37,62)(38,57)(44,64)(47,51)(49,52)(55,63)(56,60)(58,61);(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)(21,23)(22,25)(24,44)(26,45)(31,34)(32,54)(33,41)(35,37)(36,39)(38,55)(40,46)(47,49)(48,64)(50,53)(51,52)(56,58)(57,63)(59,62)(60,61);(1,5,22,52,7,24,6,23)(2,10,36,61,12,38,11,37)(3,14,25,51,15,44,16,21)(4,18,20,49,19,48,26,53)(8,28,39,60,29,55,30,35)(9,32,34,58,33,57,40,62)(13,42,17,47,43,64,45,50)(27,54,31,56,41,63,46,59)Polycyclic presentation:
2^9|a^2=i,e^2=fhi,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, 6699604);