Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #24278 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C25×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C2×#121 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C23×C4 | C24 |
| prime radical | C24×D4 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C24×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
2550 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)(3,4)(5,6,8,7)(9,10)(11,19)(12,17)(13,16)(14,18)(15,21)(20,23)(22,24);(2,4)(5,7,8,6)(9,11,14,22,12,21,13,20)(10,15,18,23,17,19,16,24);(10,16)(11,20)(13,14)(15,19)(17,18)(21,22);(2,4)(10,17)(15,19)(16,18)(23,24)Polycyclic presentation:
2^9|a^2=h,b^2=gh,f^2=i,g^2=i,b^a=be,c^a=cf,d^a=de,f^a=fi,c^b=cg,f^c=fi,g^c=gi92 conjugacy classes — view the full character table.
| maximal subgroup of | #72239135, #72239140, #72239148 |
| maximal normal subgroup of | #72239135, #72239140, #72239148 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 273501);