Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #33605 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C28 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | C22×D8 |
| commutator, Frattini, cosocle | C2×C8 | C25 |
| socle | C22 | C23×D8 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C22×C42 ◁ C22×C4×C8 ◁ G | |
| LECS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
1547 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 24 points:
(1,2)(3,6)(4,5)(7,8)(10,14)(11,15)(12,18)(16,20)(17,24)(19,23)(21,22);(2,5)(6,8);(9,10,15,22,12,17,19,20,13,16,23,24,18,21,11,14);(1,3,4,7)(2,6,5,8);(10,16)(14,20)(17,21)(22,24)Polycyclic presentation:
2^9|c^2=gh,d^2=f,g^2=hi,h^2=i,b^a=bf,c^a=cg,g^a=gh,h^a=hi,e^c=ei104 conjugacy classes — view the full character table.
| maximal subgroup of | #72225185, #72225297, #72225299, … |
| maximal normal subgroup of | #72225185, #72225297, #72225299, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7530451);