Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #428059 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C2×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #2749 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| prime radical | C24 | C2×#41 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
609 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 24 points:
(9,10,12,15,14,18,13,17)(11,19,21,16,23,20,22,24);(1,2,4,6)(3,7,8,5)(9,10)(11,20)(12,17)(13,15)(14,18)(16,21)(19,23)(22,24);(1,3,4,8)(2,5,6,7)(10,15,18,17)(16,19,24,20);(9,11,12,22,14,23,13,21)(10,16,17,20,18,24,15,19)Polycyclic presentation:
2^9|a^2=e,b^2=gh,c^2=fg,d^2=f,e^2=i,f^2=i,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,f^a=fi,c^b=cg,d^b=de,e^b=ei,g^b=gi,h^b=hi,e^d=ei62 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 361102);