Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #23860 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C25×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 | C23×D4 | C23 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ C2×C4×C4⋊C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1796 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 4 generators on 24 points:
(1,2)(3,4)(5,6,8,7)(17,18)(19,23)(20,22)(21,24);(5,7,8,6)(9,10,12,14)(11,15,16,13)(18,22)(19,20)(23,24);(2,4)(9,11,12,16)(10,13,14,15);(17,19)(20,21)(22,23)Polycyclic presentation:
2^9|a^2=h,b^2=gh,c^2=g,e^2=i,b^a=be,c^a=cf,d^a=de,e^a=ei,c^b=cg,d^b=di,e^b=ei,e^d=ei92 conjugacy classes — view the full character table.
| maximal subgroup of | #72249728, #72249746, #72249786, … |
| maximal normal subgroup of | #72249728, #72249746, #72249786, … |
| maximal quotient of | #72250817, #72265169 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 277162);