Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #7855 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C42 · Schur multiplier (H2) = C24×C42 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C42 | C2×C42 |
| commutator | C2×C4 | C22×C42 |
| Frattini, cosocle | C23×C4 | C24 |
| socle | C22 | C22×#95 |
| prime radical | C25 | C2×Q8 |
| LCS | G ▷ C2×C4 ▷ C1 | |
| UCS | C1 ◁ C42 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1826 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 24 points:
(1,2)(3,6)(4,5)(7,8)(9,10,12,16,14,17,13,22)(11,18,20,15,21,23,19,24);(9,11,13,20,14,21,12,19)(10,15,16,18,17,24,22,23);(2,5)(6,8);(1,3,4,7)(2,6,5,8)Polycyclic presentation:
2^9|a^2=g,b^2=h,d^2=f,e^2=i,g^2=i,h^2=i,b^a=be,c^a=cf,h^a=hi,g^b=gi128 conjugacy classes — view the full character table.
| maximal subgroup of | #72224690, #72224699, #72224742, … |
| maximal normal subgroup of | #72224690, #72224699, #72224742, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 260974);