Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #7563346 |
| 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 | |
1971 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)(11,19)(12,21)(13,17)(14,16)(15,20)(18,22)(23,24);(2,5)(6,8)(9,11)(10,15)(12,22)(13,20)(14,23)(16,21)(17,19)(18,24);(1,3,4,7)(2,6,5,8)(9,12,22,11,13,23,14,20)(10,15,16,24,17,19,18,21);(10,16,17,18)(11,20)(12,23)(15,21,19,24);(2,5)(6,8)(10,17)(15,19)(16,18)(21,24)Polycyclic presentation:
2^9|c^2=fg,d^2=hi,g^2=i,b^a=bf,c^a=cg,d^a=dg,e^a=ef,g^a=gi,h^a=hi,c^b=cg,d^b=dh,g^b=gi,d^c=di68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7520665);