Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #24269 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C23×C42 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C24 | C22×D4 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C23×C4 | C24 |
| prime radical | C22×#121 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ C2×C4×C22⋊C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1556 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 4 generators on 24 points:
(1,2)(3,4)(5,6,8,7)(9,10)(11,15)(12,14)(13,16)(17,18,22,23)(19,24,20,21);(5,7,8,6)(9,11,12,16)(10,13,14,15)(17,19,21,18,22,20,24,23);(2,4)(11,16)(13,15)(18,23)(19,20);(10,14)(13,15)(17,20,21,23,22,19,24,18)Polycyclic presentation:
2^9|a^2=h,b^2=ehi,d^2=egi,e^2=i,b^a=be,c^a=cf,d^a=de,e^a=ei,c^b=cg,d^c=di92 conjugacy classes — view the full character table.
| maximal subgroup of | #72264651, #72264673, #72264751, … |
| maximal normal subgroup of | #72264651, #72264673, #72264751, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 278362);