Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #60457 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C4 · Schur multiplier (H2) = C29×C4 |
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 | C22×#576 | C2 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C2×C4×#41 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
5292 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 24 points:
(2,4)(5,6,7,8)(10,11)(13,14)(16,19)(17,18)(21,22)(23,24);(1,2,3,4)(6,8)(9,10,14,17,15,18,13,11)(12,16,22,24,20,23,21,19);(9,11,13,18,15,17,14,10)(12,19,21,23,20,24,22,16);(9,12)(10,16)(11,19)(13,21)(14,22)(15,20)(17,24)(18,23);(12,20)(16,23)(19,24)(21,22)Polycyclic presentation:
2^9|a^2=h,b^2=fhi,c^2=gi,f^2=i,g^2=i,b^a=bf,c^a=cg,f^a=fi,g^a=gi,e^d=ei128 conjugacy classes — view the full character table.
| maximal subgroup of | C22⋊C4×#576, #72284953, #72285004, … |
| maximal normal subgroup of | C22⋊C4×#576, #72284953, #72285004, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6279204);