Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6682648 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C28 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C2×D42 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1660 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 5 generators on 48 points:
(1,2)(3,9)(4,8)(5,7)(6,11)(10,16)(12,14)(13,15)(17,18)(19,29)(20,21)(22,27)(23,28)(24,26)(25,32)(30,33)(31,45)(34,36)(35,41)(37,48)(38,44)(39,42)(40,46)(43,47);(1,3,5,11)(2,6,7,9)(4,12,13,10)(8,16,15,14)(17,19,23,32)(18,25,28,29)(20,33,26,41)(21,35,24,30)(22,37,38,47)(27,43,44,48)(31,39,46,36)(34,40,42,45);(17,20,21,18,23,26,24,28)(19,29,35,33,32,25,30,41)(22,34,36,27,38,42,39,44)(31,45,47,48,46,40,37,43);(2,7)(6,9)(8,15)(14,16)(17,21,23,24)(18,26,28,20)(19,30,32,35)(22,36,38,39)(25,33,29,41)(27,42,44,34)(31,37,46,47)(40,48,45,43);(1,4,5,13)(2,8,7,15)(3,10,11,12)(6,14,9,16)(17,22,23,38)(18,27,28,44)(19,31,32,46)(20,34,26,42)(21,36,24,39)(25,40,29,45)(30,37,35,47)(33,48,41,43)Polycyclic presentation:
2^9|b^2=f,c^2=gi,d^2=i,e^2=f,g^2=i,h^2=i,b^a=bf,c^a=cg,d^a=df,g^a=gi,c^b=ch,d^b=di,e^b=efh,h^b=hi,h^c=hi,h^e=hi62 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6593306);