Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #348108 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C25×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #2847 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| LCS | G ▷ C23×C4 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C22 ▷ C1 | |
1260 conjugacy classes of subgroups; only the 162 normal subgroups are shown.
This group has 1260 conjugacy classes of subgroups — too many to draw. Only its 162 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) = 64 |
A minimal faithful permutation representation — 4 generators on 24 points:
(5,8)(10,14)(12,15)(16,20)(17,21)(19,22)(23,24);(9,10,15,21,13,17,12,14)(11,16,22,24,18,23,19,20);(1,2)(3,5)(4,6)(7,8)(9,11)(10,16)(12,19)(13,18)(14,20)(15,22)(17,23)(21,24);(1,3)(2,5)(4,7)(6,8)(11,18)(16,23)(19,22)(20,24)Polycyclic presentation:
2^9|b^2=eh,e^2=h,b^a=be,c^a=cf,d^a=dg,e^a=eh,d^c=dh,g^c=gi,f^d=fi59 conjugacy classes — view the full character table.
| maximal subgroup of | #72268350, #72268356, #72268370, … |
| maximal normal subgroup of | #72268350, #72268356, #72268370, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 255305);