Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #61049 |
| 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 | C22×#493 | C2 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2285 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
3270 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)(5,6,8,7)(9,10,14,17,15,11,13,19)(12,16,22,24,23,18,21,20);(5,7,8,6)(9,11)(10,15)(12,20)(13,19)(14,17)(16,21)(18,22)(23,24);(1,2,3,4);(9,12,13,21,15,23,14,22)(10,16,19,20,11,18,17,24);(9,13,15,14)(10,17,11,19)(12,21,23,22)(16,24,18,20)Polycyclic presentation:
2^9|a^2=h,b^2=fhi,c^2=g,d^2=fi,e^2=i,f^2=i,h^2=i,b^a=bf,c^a=cg,e^a=ei,d^b=df,f^b=fi,h^b=hi116 conjugacy classes — view the full character table.
| maximal subgroup of | #72214822, #72280417, #72311786, … |
| maximal normal subgroup of | #72214822, #72280417, #72311786, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6289968);