Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #17392 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C42 · Schur multiplier (H2) = C24×C42 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | C22×C22⋊C4 |
| commutator | C23 | C22×C42 |
| Frattini, cosocle | C23×C4 | C24 |
| socle | C22 | C22×#95 |
| prime radical | C25 | C22×C4 |
| LCS | G ▷ C23 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C23×C42 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1770 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 24 points:
(1,2)(3,7)(4,6)(5,8)(9,10)(11,14,17,21)(12,19)(13,15)(16,20)(18,24,23,22);(1,3,4,8)(2,5,6,7);(3,8)(5,7)(9,11,12,18,13,17,20,23)(10,14,16,22,15,21,19,24);(2,6)(5,7)Polycyclic presentation:
2^9|a^2=g,b^2=e,c^2=h,h^2=i,b^a=be,c^a=cf,d^a=de,f^a=fi,h^a=hi,c^b=ce,f^c=fi,g^c=gi104 conjugacy classes — view the full character table.
| maximal subgroup of | #72311070, #72311074, #72311197, … |
| maximal normal subgroup of | #72311070, #72311074, #72311197, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 263200);