Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #301993 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C25×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #2433 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| LCS | G ▷ C23×C4 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C22 ▷ C1 | |
1254 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 24 points:
(1,2)(3,7)(4,6)(5,8)(9,10)(11,19)(12,18)(13,17)(14,16)(15,20)(21,24)(22,23);(1,3,4,8)(2,5,6,7)(9,10,14,18,13,17,12,16)(11,15,22,24,21,23,20,19);(1,2)(3,7,8,5)(4,6)(9,10,13,17)(11,15,21,23)(12,16,14,18)(19,22,24,20);(2,6)(5,7)(9,11,12,20,13,21,14,22)(10,15,16,19,17,23,18,24)Polycyclic presentation:
2^9|b^2=eh,c^2=f,d^2=ei,e^2=h,g^2=h,b^a=be,c^a=cf,d^a=de,e^a=eh,g^a=gi,c^b=cg,f^b=fi,g^b=gh,d^c=di,g^c=ghi53 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 209207);