Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #133875 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C25×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #1781 |
| commutator | C23×C4 | C22×C4 |
| Frattini, cosocle | C23×D4 | C23 |
| prime radical | C22×#127 | C22 |
| LCS | G ▷ C23×C4 ▷ C23 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C24 ◁ C22×#124 ◁ G | |
| LECS | G ▷ C23×D4 ▷ C23 ▷ C2 ▷ C1 | |
1330 conjugacy classes of subgroups; only the 94 normal subgroups are shown.
This group has 1330 conjugacy classes of subgroups — too many to draw. Only its 94 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) = 32 |
A minimal faithful permutation representation — 3 generators on 24 points:
(1,2)(3,7,8,5)(4,6)(9,10)(11,18,20,15)(12,21,22,16)(13,17)(14,19)(23,24);(9,11)(10,14,17,23)(12,19)(13,20)(15,16,18,21)(22,24);(1,3,4,8)(2,5,6,7)(9,11)(10,15)(12,19)(13,20)(14,16)(17,18)(21,23)(22,24)Polycyclic presentation:
2^9|a^2=f,b^2=g,c^2=hi,d^2=i,b^a=bd,c^a=ce,d^a=dg,e^a=eh,g^a=gi,f^b=fgi,f^c=fh,f^d=fi56 conjugacy classes — view the full character table.
| maximal subgroup of | #72341386, #72341424, #72341498, … |
| maximal normal subgroup of | #72341386, #72341424, #72341498, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 41206);