Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #1999 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C42 · Schur multiplier (H2) = C29 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C25 | C2×D4 |
| commutator | C22×C4 | C2×C42 |
| Frattini, cosocle | C24×C4 | C23 |
| prime radical | C24×D4 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C25 ◁ C23×C42 ◁ G | |
| LECS | G ▷ C24×C4 ▷ C2 ▷ C1 | |
3894 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 3 generators on 24 points:
(1,2)(3,4)(9,10,11,12)(17,18)(19,22)(20,23)(21,24);(2,4)(5,6)(7,8)(13,14,15,16)(18,22)(19,20)(23,24);(6,8)(18,23)(19,20)(22,24)Polycyclic presentation:
2^9|a^2=g,b^2=h,d^2=i,e^2=i,b^a=bd,c^a=ce,d^a=di,e^a=ei,c^b=cf,d^b=di,e^b=ei,d^c=di,e^c=ei128 conjugacy classes — view the full character table.
| maximal subgroup of | #72216608, #72314205, #72317054, … |
| maximal normal subgroup of | #72216608, #72314205, #72317054, … |
| maximal quotient of | #72216508, #72216512, #72216515, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 30760);