Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #60367 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C4 · Schur multiplier (H2) = C210 |
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×#3013 | C2 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C2×C4×#41 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
4590 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)(9,10,14,18)(11,19,21,23)(12,17,13,16)(15,22,24,20);(9,11,14,21)(10,15,18,24)(12,20,13,22)(16,19,17,23);(2,4)(9,10)(11,15)(12,16)(13,17)(14,18)(19,20)(21,24)(22,23);(5,7,8,6)(9,12,14,13)(10,16,18,17)(11,20,21,22)(15,19,24,23);(9,12,14,13)(10,17,18,16)(11,20,21,22)(15,23,24,19)Polycyclic presentation:
2^9|a^2=h,b^2=i,d^2=h,e^2=i,f^2=i,b^a=bf,c^a=cg,d^a=di,f^a=fi,f^b=fi,e^c=ei116 conjugacy classes — view the full character table.
| maximal subgroup of | #72285158, #72285294 |
| maximal normal subgroup of | #72285158, #72285294 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6279856);