Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #60457 |
| 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 | C23×#41 | C22 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C23×#41 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
4770 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 24 points:
(1,2)(3,4)(5,6,7,8)(10,14)(12,15)(16,20)(17,21)(19,22)(23,24);(9,10,15,21,13,17,12,14)(11,16,22,24,18,23,19,20);(2,4);(9,11)(10,16)(12,19)(13,18)(14,20)(15,22)(17,23)(21,24);(11,18)(16,23)(19,22)(20,24)Polycyclic presentation:
2^9|a^2=h,b^2=fi,f^2=i,b^a=bf,c^a=cg,f^a=fi,e^d=ei128 conjugacy classes — view the full character table.
| maximal subgroup of | C22⋊C4×#576, #72284669 |
| maximal normal subgroup of | C22⋊C4×#576, #72284669 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6278047);