Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6489185 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C27×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #35237 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C22×C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1709 conjugacy classes of subgroups.
| minimal degree | μ(G) = 64 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 5 generators on 64 points:
(1,2,6,13,8,14,7,10)(3,15,16,30,20,33,19,9)(4,11,23,37,25,38,24,34)(5,26,27,40,28,39,29,12)(17,41,43,54,46,35,21,31)(18,32,47,57,48,55,44,42)(22,50,51,60,52,59,53,36)(45,56,49,58,63,64,62,61);(1,3)(2,9)(4,21)(5,18)(6,19)(7,16)(8,20)(10,15)(11,35)(12,32)(13,33)(14,30)(17,24)(22,49)(23,46)(25,43)(26,42)(27,44)(28,48)(29,47)(31,34)(36,58)(37,54)(38,41)(39,57)(40,55)(45,51)(50,56)(52,62)(53,63)(59,64)(60,61);(2,10,14,13)(3,16,20,19)(9,30)(11,34,38,37)(12,26,40,39)(15,33)(17,43,46,21)(18,44,48,47)(31,54)(32,55)(35,41)(36,50,60,59)(42,57)(45,62,63,49)(56,64)(58,61);(1,4,7,24,8,25,6,23)(2,11,10,34,14,38,13,37)(3,17,19,21,20,46,16,43)(5,22,27,51,28,52,29,53)(9,31,33,35,30,54,15,41)(12,36,26,50,40,60,39,59)(18,45,47,49,48,63,44,62)(32,56,57,58,55,64,42,61);(1,5,7,27,8,28,6,29)(2,12,13,39,14,40,10,26)(3,18,16,44,20,48,19,47)(4,22,24,51,25,52,23,53)(9,32,33,57,30,55,15,42)(11,36,37,59,38,60,34,50)(17,45,43,62,46,63,21,49)(31,56,35,58,54,64,41,61)Polycyclic presentation:
2^9|a^2=f,c^2=ghi,d^2=h,e^2=g,f^2=i,g^2=i,h^2=i,b^a=bf,c^a=cg,e^a=eh,g^a=gi,c^b=ch,d^b=dh,f^b=fi,h^b=hi,f^e=fi68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6396382);