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×#3271 | C2 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2290 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2675 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 24 points:
(1,2)(3,4)(5,6,8,7)(9,10)(11,20)(12,19)(13,21)(14,16)(15,18)(17,23)(22,24);(9,11,15,22)(10,13,18,23)(12,21,14,17)(16,20,19,24);(2,4)(9,12,15,14)(10,16,18,19)(11,21,22,17)(13,20,23,24);(9,13,15,23)(10,17,18,21)(11,19,22,16)(12,20,14,24);(5,7,8,6)(10,18)(11,22)(16,19)(17,21)Polycyclic presentation:
2^9|a^2=h,b^2=i,c^2=i,d^2=i,e^2=h,f^2=i,b^a=bf,c^a=cg,d^a=di,e^a=ei,f^a=fi,d^b=df,e^b=ei,f^b=fi,f^d=fi110 conjugacy classes — view the full character table.
| maximal subgroup of | #72284355, #72284895, #72285294 |
| maximal normal subgroup of | #72284355, #72284895, #72285294 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6291178);