Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #125068 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C22×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C2×#99 |
| commutator | C23×C4 | C22×C4 |
| Frattini, cosocle | C2×#95 | C23 |
| prime radical | C24 | C2×C22⋊C4 |
| LCS | G ▷ C23×C4 ▷ C23 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C23×C4 ◁ C2×C4×C4⋊C4 ◁ G | |
| LECS | G ▷ C2×#95 ▷ C24 ▷ C2 ▷ C1 | |
610 conjugacy classes of subgroups; only the 108 normal subgroups are shown.
This group has 610 conjugacy classes of subgroups — too many to draw. Only its 108 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 3 generators on 24 points:
(2,5,6,7)(9,10,12,14,13,16,11,15)(17,18,20,24,21,23,19,22);(1,2)(3,7)(4,6)(5,8)(10,14)(11,12)(15,16)(17,19,21,20)(18,22)(23,24);(9,11,13,12)(10,15,16,14)Polycyclic presentation:
2^9|a^2=e,b^2=fi,c^2=hi,d^2=gi,e^2=h,b^a=bd,d^a=df,f^a=fi,c^b=chi,d^b=dg,e^b=efgi,e^d=ei68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 32388);