Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #2846 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C4 · Schur multiplier (H2) = C29 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C22×C8 | C24 |
| commutator, socle | C23 | C24×C4 |
| Frattini, cosocle | C22×C4 | C25 |
| prime radical | #36624 | C2 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C22×C8 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
3198 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 24 points:
(2,4)(5,6)(7,8)(9,10,12,16,13,15,19,22)(11,14,17,21,18,20,23,24);(1,2,3,4);(6,8);(9,11,12,17,13,18,19,23)(10,14,16,21,15,20,22,24);(10,15)(14,20)(16,22)(21,24)Polycyclic presentation:
2^9|a^2=h,b^2=f,d^2=h,h^2=i,b^a=bf,c^a=cg,e^a=ei176 conjugacy classes — view the full character table.
| maximal subgroup of | #72286025, #72286144, #72309912, … |
| maximal normal subgroup of | #72286025, #72286144, #72309912, … |
| maximal quotient of | #72286144, #72286217, C8×#2847, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6279957);