Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6682638 |
| 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 | |
1830 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 5 generators on 24 points:
(1,2)(3,6)(4,5)(7,8)(9,10)(11,16)(12,22)(13,24)(14,18)(15,19)(17,20)(21,23);(2,5)(3,7)(9,11)(10,16)(12,13)(14,23)(15,20)(17,19)(18,21)(22,24);(9,12,14,11,15,23,13,20)(10,17,24,21,19,16,18,22);(2,5)(6,8)(11,20)(12,23)(16,17)(21,22);(1,3,4,7)(2,6,5,8)(9,10)(11,21)(12,17)(13,18)(14,24)(15,19)(16,23)(20,22)Polycyclic presentation:
2^9|c^2=gi,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,d^c=di,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, 6592720);