Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #24378 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C26×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 | C25 | C2×D4 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ C23×C42 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
2042 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 4 generators on 24 points:
(2,4)(9,10)(11,15)(12,14)(13,16)(17,18,19,22,20,23,24,21);(5,6)(7,8)(9,11,13,12)(10,14,16,15);(1,2,3,4)(6,8)(9,11,13,12)(10,15,16,14)(18,21)(19,24)(22,23);(9,12,13,11)(10,15,16,14)Polycyclic presentation:
2^9|a^2=h,b^2=e,c^2=efhi,d^2=e,f^2=i,h^2=i,b^a=be,c^a=cf,d^a=de,c^b=cg,f^c=fi,h^c=hi116 conjugacy classes — view the full character table.
| maximal subgroup of | #72304900, #72304903, #72304907, … |
| maximal normal subgroup of | #72304900, #72304903, #72304907, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 281811);