Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #480460 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C24×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C22×#99 |
| commutator | C24 | C23×C4 |
| Frattini, cosocle | C22×D4 | C24 |
| prime radical | #2403 | C22 |
| LCS | G ▷ C24 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C2×C22⋊C4 ◁ C2×#518 ◁ G | |
| LECS | G ▷ C22×D4 ▷ C22 ▷ C2 ▷ C1 | |
1150 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 24 points:
(1,2)(3,6)(4,5)(7,8)(9,10,12,17,13,16,14,20)(11,15,19,23,18,22,21,24);(9,10,14,17)(11,15,21,23)(12,20,13,16)(18,22,19,24);(2,5)(6,8)(9,11)(10,15)(12,19)(13,18)(14,21)(16,22)(17,23)(20,24);(1,3,4,7)(2,6,5,8)(11,18)(15,22)(19,21)(23,24)Polycyclic presentation:
2^9|a^2=g,b^2=egi,d^2=f,g^2=i,b^a=be,c^a=cf,e^a=eh,h^a=hi,e^b=eh,g^b=gh,h^b=hi,d^c=di,g^e=gi68 conjugacy classes — view the full character table.
| maximal subgroup of | #72318261, #72318299, #72318328, … |
| maximal normal subgroup of | #72318261, #72318299, #72318328, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 402831);