Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #465 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C42 · Schur multiplier (H2) = C25×C42 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C23×C4 | C2×D4 |
| commutator | C2×C4 | C22×C42 |
| Frattini, cosocle | C23×C4 | C24 |
| socle | C24 | C22×D4 |
| prime radical | C25 | C2×D4 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23×C4 ◁ C23×C42 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
2522 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)(5,6,7,8)(9,10)(11,14)(12,13)(15,16)(18,22)(19,20)(23,24);(2,4)(17,18,20,23,21,24,19,22);(10,13)(14,16);(9,11,12,15)(10,14,13,16)Polycyclic presentation:
2^9|a^2=g,b^2=h,d^2=f,e^2=i,h^2=i,b^a=be,c^a=cf,e^a=ei,h^a=hi176 conjugacy classes — view the full character table.
| maximal subgroup of | #72321653, #72321661, #72321675, … |
| maximal normal subgroup of | #72321653, #72321661, #72321675, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 260981);