Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #7535051 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C28 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #59041 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C22×C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1742 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 5 generators on 32 points:
(2,9)(3,15)(4,19)(5,18)(6,7)(10,12)(11,26)(13,14)(17,31)(22,23)(25,32)(27,30);(1,2,6,9,8,14,7,13)(3,16,5,12,18,24,15,10)(4,20,22,26,23,28,19,11)(17,27,21,32,31,25,29,30);(1,3,7,15,8,18,6,5)(2,10,9,16,14,12,13,24)(4,17,19,29,23,31,22,21)(11,25,20,30,26,27,28,32);(1,4,6,22,8,23,7,19)(2,11,9,20,14,26,13,28)(3,17,5,21,18,31,15,29)(10,25,16,30,12,27,24,32);(1,5,6,18,8,15,7,3)(2,12,9,24,14,10,13,16)(4,21,22,31,23,29,19,17)(11,27,20,32,26,25,28,30)Polycyclic presentation:
2^9|b^2=f,c^2=g,d^2=f,e^2=f,f^2=i,g^2=i,h^2=i,b^a=bf,c^a=cg,d^a=dg,e^a=ef,f^a=fi,g^a=gi,h^a=hi,c^b=cg,d^b=dh,g^b=gi,h^b=hi,h^d=hi68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7442247);