Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #24269 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C23×C42 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C22×C4 | C22×D4 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C23×C4 | C24 |
| socle | C23 | C2×C4×D4 |
| prime radical | C2×#532 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C4 ◁ C22×C4⋊C8 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1214 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 4 generators on 24 points:
(2,3)(6,8)(9,10,13,18,14,16,12,17)(11,15,19,22,20,23,21,24);(1,2,4,3)(11,19,20,21)(15,22,23,24);(5,6,7,8)(9,11)(10,15)(12,21)(13,19)(14,20)(16,23)(17,24)(18,22);(1,3,4,2)(10,16)(15,23)(17,18)(22,24)Polycyclic presentation:
2^9|a^2=h,b^2=egh,c^2=f,d^2=e,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=dei,c^b=cg,g^c=gi92 conjugacy classes — view the full character table.
| maximal subgroup of | #72310050, D4×#1938, #72310141 |
| maximal normal subgroup of | #72310050, D4×#1938, #72310141 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 274469);