Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #30412 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C25×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C24 | C22×D4 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C23×C4 | C24 |
| prime radical | C23×D8 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ C2×C4×C4⋊C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1614 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 4 generators on 24 points:
(2,3)(5,6,7,8)(9,10)(11,13)(12,15)(14,16)(18,21)(22,23);(9,11)(10,12)(13,16)(14,15)(17,18)(19,21)(20,22)(23,24);(1,2,4,3)(9,12,14,13)(10,15,16,11)(18,22)(20,24)(21,23);(1,3,4,2)(9,13,14,12)(10,15,16,11)(18,21)(22,23)Polycyclic presentation:
2^9|a^2=h,c^2=efi,d^2=efi,g^2=i,b^a=be,c^a=cf,d^a=defi,c^b=cg,d^b=de,g^b=gi,g^c=gi80 conjugacy classes — view the full character table.
| maximal subgroup of | #72239681, #72239718, #72239744, … |
| maximal normal subgroup of | #72239681, #72239718, #72239744, … |
| maximal quotient of | #72250502 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 292692);