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 | |
1784 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)(3,5)(4,18)(6,7)(10,23)(11,26)(12,13)(15,29)(16,17)(21,22)(24,32)(28,30);(1,2,6,9,8,13,7,12)(3,14,16,23,17,25,5,10)(4,19,21,26,22,27,18,11)(15,30,31,32,29,24,20,28);(1,3,6,16,8,17,7,5)(2,10,12,25,13,23,9,14)(4,15,21,31,22,29,18,20)(11,24,27,32,26,30,19,28);(1,4,6,21,8,22,7,18)(2,11,9,19,13,26,12,27)(3,15,16,31,17,29,5,20)(10,24,14,28,23,30,25,32);(1,5,7,17,8,16,6,3)(2,10,12,25,13,23,9,14)(4,20,18,29,22,31,21,15)(11,24,27,32,26,30,19,28)Polycyclic presentation:
2^9|b^2=f,c^2=gi,d^2=f,e^2=fi,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, 7442249);