Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6500006 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C29 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C22×#120 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C22×#41 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
3235 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 5 generators on 24 points:
(10,14,16,20)(11,17)(15,22,23,21)(18,24);(1,2)(3,7)(4,6)(5,8)(12,19)(14,20)(18,24)(21,22);(1,3)(2,5)(4,8)(6,7)(9,10)(11,15)(12,20)(13,16)(14,19)(17,23)(18,22)(21,24);(3,8)(5,7)(9,11,13,17)(10,15,16,23)(12,18,19,24)(14,21,20,22);(2,6)(5,7)(11,17)(15,23)(18,24)(21,22)Polycyclic presentation:
2^9|a^2=f,d^2=i,g^2=i,b^a=bf,c^a=cg,d^a=di,c^b=ch,e^b=eh,g^b=gi,d^c=dh,f^c=fi,g^c=gi,e^d=ei68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6407204);