Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #24808 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C29 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C4 | #2416 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| socle | C2 | C2×#2416 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2445 conjugacy classes of subgroups.
| minimal degree | μ(G) = 64 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 5 generators on 64 points:
(1,2,6,12,8,14,7,13)(3,15,19,34,20,33,18,9)(4,21,24,10,25,37,29,38)(5,11,27,40,28,42,30,41)(16,43,22,50,46,31,45,35)(17,39,26,55,48,57,47,32)(23,49,52,36,54,59,53,60)(44,61,51,64,63,56,62,58);(1,3)(2,9)(4,22)(5,26)(6,18)(7,19)(8,20)(10,35)(11,39)(12,33)(13,15)(14,34)(16,24)(17,27)(21,43)(23,44)(25,45)(28,47)(29,46)(30,48)(31,37)(32,40)(36,56)(38,50)(41,55)(42,57)(49,58)(51,53)(52,62)(54,63)(59,64)(60,61);(1,4,6,24,8,25,7,29)(2,10,13,21,14,38,12,37)(3,16,18,45,20,46,19,22)(5,23,27,52,28,54,30,53)(9,31,15,35,34,43,33,50)(11,36,41,49,42,60,40,59)(17,44,47,62,48,63,26,51)(32,56,39,58,55,61,57,64);(1,5)(2,11)(3,17)(4,23)(6,27)(7,30)(8,28)(9,32)(10,36)(12,40)(13,41)(14,42)(15,39)(16,44)(18,47)(19,26)(20,48)(21,49)(22,51)(24,52)(25,54)(29,53)(31,56)(33,57)(34,55)(35,58)(37,59)(38,60)(43,61)(45,62)(46,63)(50,64);(1,6,8,7)(2,12,14,13)(3,18,20,19)(4,24,25,29)(5,27,28,30)(9,33,34,15)(10,37,38,21)(11,40,42,41)(16,45,46,22)(17,47,48,26)(23,52,54,53)(31,50,43,35)(32,57,55,39)(36,59,60,49)(44,62,63,51)(56,64,61,58)Polycyclic presentation:
2^9|a^2=f,c^2=g,e^2=i,f^2=i,g^2=i,h^2=i,b^a=bf,c^a=cg,g^a=gi,c^b=ch,d^b=dh,f^b=fi,h^b=hi,h^c=hi,h^d=hi80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6350256);