Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #469776 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C22×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #2631 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| prime radical | C24 | H(5,𝔽2) |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
740 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)(11,18,21,24)(12,15,20,23)(13,16)(14,17)(19,22);(1,2,4,6)(3,7,8,5)(9,11,19,12,14,21,13,20)(10,15,16,18,17,23,22,24);(1,3,4,8)(2,5,6,7)(9,10)(11,15)(12,18)(13,22)(14,17)(16,19)(20,24)(21,23);(9,12,13,11,14,20,19,21)(10,15,16,18,17,23,22,24)Polycyclic presentation:
2^9|a^2=f,b^2=egh,c^2=g,d^2=e,e^2=i,b^a=be,c^a=cf,d^a=dh,h^a=hi,c^b=cg,f^b=fi,d^c=de,e^c=ei,h^c=hi,f^d=fi56 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 381821);