Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #25140 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C25 |
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 | C2×#41 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C22×C42 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
924 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 4 generators on 24 points:
(2,3)(5,6,8,7)(9,10)(11,18)(12,21)(13,16)(14,23)(15,20)(17,19)(22,24);(1,2,4,3)(5,7,8,6)(9,11,19,12,13,20,14,22)(10,15,17,24,16,18,23,21);(1,2,4,3)(9,12,14,20,13,22,19,11)(10,15,17,21,16,18,23,24);(1,3,4,2)(10,16)(15,18)(17,23)(21,24)Polycyclic presentation:
2^9|a^2=h,b^2=fgh,c^2=f,d^2=ei,f^2=i,b^a=be,c^a=cf,d^a=de,f^a=fi,g^a=gi,c^b=cg,f^b=fi,g^b=gi,g^c=gi86 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 274734);