Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #24384 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C24×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C22×C4 | C22×D4 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C2×C42 | C24 |
| socle | C23 | C22×C22⋊C4 |
| prime radical | C24 | C2×C22⋊C4 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C4 ◁ C2×C43 ◁ G | |
| LECS | G ▷ C2×C42 ▷ C22 ▷ C1 | |
1210 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)(3,7)(4,5)(6,8)(9,10)(11,15)(12,16)(13,14)(17,18,20,23,21,19,22,24);(2,5)(6,7)(9,11,15,10,13,16,12,14)(17,19,20,24,21,18,22,23);(1,3,4,8)(2,6,5,7)(10,14)(11,16);(18,19)(23,24)Polycyclic presentation:
2^9|a^2=g,b^2=efg,c^2=fh,e^2=h,g^2=i,b^a=be,c^a=cf,d^a=di,e^a=eh,c^b=ch,d^b=di92 conjugacy classes — view the full character table.
| maximal subgroup of | #72238780, #72239578, #72257948, … |
| maximal normal subgroup of | #72238780, #72239578, #72257948, … |
| maximal quotient of | #72237291 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 114034);