Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #188761 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C25×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | C2×#117 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C23×C4 | C24 |
| socle | C22 | C2×#381 |
| prime radical | C25 | C2×D4 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C22×#117 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1542 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 4 generators on 24 points:
(1,2)(3,5)(4,6)(7,8)(9,10)(11,18,20,14)(12,17)(13,16)(15,19,22,24)(21,23);(1,3,4,7)(2,5,6,8)(9,11)(10,14)(12,19)(13,20)(15,17)(16,18)(21,24)(22,23);(2,6)(5,8);(9,11)(10,15,16,22)(12,19)(13,20)(14,17,18,23)(21,24)Polycyclic presentation:
2^9|a^2=h,b^2=f,d^2=e,g^2=i,b^a=be,c^a=cf,d^a=dg,e^a=ei,h^b=hi,h^d=hi104 conjugacy classes — view the full character table.
| maximal subgroup of | #72240734, #72240804, #72240805, … |
| maximal normal subgroup of | #72240734, #72240804, #72240805, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 299570);