Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #500834 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C25×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C22×D8 |
| commutator, Frattini, cosocle | C22×C8 | C24 |
| LCS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C2×C4×D4 ◁ C2×C8×D4 ◁ G | |
| LECS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
1173 conjugacy classes of subgroups; only the 178 normal subgroups are shown.
This group has 1173 conjugacy classes of subgroups — too many to draw. Only its 178 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 — 4 generators on 24 points:
(1,2)(3,4)(10,14)(11,15)(13,18)(16,20)(17,24)(19,23)(21,22);(5,6)(7,8)(9,10,15,22,13,17,19,20,12,16,23,24,18,21,11,14);(2,4)(6,8)(10,16)(14,20)(17,21)(22,24);(9,10,15,22,13,17,19,20,12,16,23,24,18,21,11,14)Polycyclic presentation:
2^9|b^2=eh,d^2=eh,e^2=hi,h^2=i,b^a=be,c^a=cf,d^a=de,e^a=eh,h^a=hi,c^b=cg,d^c=di74 conjugacy classes — view the full character table.
| maximal subgroup of | #72318484, #72319173, #72319560, … |
| maximal normal subgroup of | #72318484, #72319173, #72319560, … |
| maximal quotient of | #72322621 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 408030);