Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #124648 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C22 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | #418 |
| commutator | C22×C8 | C22×C4 |
| Frattini, cosocle | C2×C4×C8 | C23 |
| socle | C22 | #1490 |
| prime radical | C24 | #104 |
| LCS | G ▷ C22×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C22×C8 ◁ C2×C4×M16 ◁ G | |
| LECS | G ▷ C2×C4×C8 ▷ C22×C4 ▷ C2 ▷ C1 | |
396 conjugacy classes of subgroups; only the 84 normal subgroups are shown.
This group has 396 conjugacy classes of subgroups — too many to draw. Only its 84 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)(10,14,15,23,17,22,16,18);(1,3,4,8)(2,5,6,7)(9,10)(11,18)(12,15)(13,17)(14,21)(16,20)(19,23)(22,24);(11,19)(14,22)(18,23)(21,24)Polycyclic presentation:
2^9|a^2=e,b^2=f,d^2=gi,e^2=h,g^2=i,b^a=bd,c^a=ch,d^a=df,d^b=dg,e^b=efg,g^b=gi,h^b=hi,d^c=di68 conjugacy classes — view the full character table.
| maximal subgroup of | #72219598, #72226641, #72226642, … |
| maximal normal subgroup of | #72219598, #72226641, #72226642, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 31864);