Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #32898 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C26×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C2×#120 |
| commutator | C24 | C23×C4 |
| Frattini, cosocle | C25 | C24 |
| prime radical | C23×D4 | C23 |
| LCS | G ▷ C24 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C24×C4 ◁ G | |
| LECS | G ▷ C25 ▷ C2 ▷ C1 | |
2224 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 4 generators on 24 points:
(2,4)(6,8)(9,10)(11,14)(12,19,20,15)(13,16)(17,21,23,24)(18,22);(1,2,3,4)(9,11,13,18)(10,14,16,22)(12,17,20,23)(15,21,19,24);(5,6,7,8);(9,12)(10,15)(11,17)(13,20)(14,21)(16,19)(18,23)(22,24)Polycyclic presentation:
2^9|a^2=h,b^2=ehi,c^2=f,b^a=be,c^a=cf,d^a=dg,g^a=gi,g^b=gi,e^d=ei,h^d=hi98 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 300252);