Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #24384 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C24×C4 |
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 | C22×#135 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C4 ◁ C22×C4×C8 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1185 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)(10,14,16,19)(11,17)(15,22,23,21)(18,24);(1,2,4,3)(5,6)(7,8);(6,8)(9,10)(11,15)(12,19)(13,16)(14,20)(17,23)(18,22)(21,24);(1,3,4,2)(9,11,12,18,13,17,20,24)(10,15,19,22,16,23,14,21)Polycyclic presentation:
2^9|a^2=h,b^2=e,d^2=efh,f^2=i,b^a=be,c^a=cf,d^a=dei,c^b=cg,f^c=fi,h^c=hi92 conjugacy classes — view the full character table.
| maximal subgroup of | #72309778, #72310129, #72310146 |
| maximal normal subgroup of | #72309778, #72310129, #72310146 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 284913);