Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #61071 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C4 · Schur multiplier (H2) = C29 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C23×D4 |
| commutator | C2×C4 | C24×C4 |
| Frattini, cosocle | C22×C4 | C25 |
| prime radical | C2×#532 | C22 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C2×#532 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2690 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 24 points:
(2,4)(5,6,7,8)(11,17)(12,18)(14,20)(16,21)(19,23)(22,24);(1,2)(3,4)(9,10)(11,14)(12,16)(13,15)(17,20)(18,21)(19,22)(23,24);(9,11,18,23,13,19,12,17)(10,14,21,24,15,22,16,20);(2,4)(9,10)(11,14)(12,16)(13,15)(17,20)(18,21)(19,22)(23,24);(10,15)(11,19)(16,21)(17,23)Polycyclic presentation:
2^9|a^2=h,c^2=gi,g^2=i,b^a=bf,c^a=cg,g^a=gi,d^b=df,e^b=ei,e^c=ei,e^d=ei110 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6287290);