Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #2403 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C8 · Schur multiplier (H2) = C26 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, Frattini, cosocle | C22×C8 | C24 |
| commutator, socle | C23 | C23×C8 |
| prime radical | C23×D4 | C8 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C22×C8 ◁ G | |
| LECS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
1334 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 4 generators on 24 points:
(2,4)(5,6)(7,8)(9,10,12,15,13,16,19,22,11,14,17,20,18,21,23,24);(1,2)(3,4)(10,14)(15,20)(16,21)(22,24);(6,8);(2,4)(10,14)(15,20)(16,21)(22,24)Polycyclic presentation:
2^9|a^2=g,g^2=h,h^2=i,b^a=be,c^a=cf,d^a=di,d^b=dei152 conjugacy classes — view the full character table.
| maximal subgroup of | D4×#2112, #72310193, #72310196 |
| maximal normal subgroup of | D4×#2112, #72310193, #72310196 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 404464);