Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #500839 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C25 |
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 ◁ C4×C22⋊C4 ◁ C8×C22⋊C4 ◁ G | |
| LECS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
773 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 4 generators on 24 points:
(5,6,8,7)(10,15)(11,18)(12,16)(13,21)(17,24)(19,22)(20,23);(1,2)(3,4)(5,7,8,6)(9,10,16,23,13,17,22,18,14,11,19,24,21,20,12,15);(2,4)(6,7);(9,11,19,23,13,20,12,18,14,10,16,24,21,17,22,15)Polycyclic presentation:
2^9|a^2=f,b^2=efh,d^2=ehi,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^b=di74 conjugacy classes — view the full character table.
| maximal subgroup of | #72319014, #72319493, #72319600 |
| maximal normal subgroup of | #72319014, #72319493, #72319600 |
| maximal quotient of | #72322609 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 409694);