Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #127249 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C42 · Schur multiplier (H2) = C24 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C2×#340 |
| commutator | C2×C8 | C2×C42 |
| Frattini, cosocle | C23×C8 | C23 |
| prime radical | C25 | SD16 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C24×C4 ◁ C24×C8 ◁ G | |
| LECS | G ▷ C23×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
718 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 3 generators on 24 points:
(1,2)(3,7,8,5)(4,6)(10,15,16,17)(11,19)(12,21)(13,20)(18,24,22,23);(1,3)(2,5)(4,8)(6,7)(9,10,12,17,13,18,21,24,14,16,11,15,20,22,19,23);(10,16)(15,17)(18,22)(23,24)Polycyclic presentation:
2^9|a^2=e,b^2=f,d^2=ghi,f^2=ghi,g^2=i,b^a=bd,d^a=dg,f^a=fghi,g^a=gi,c^b=ci,e^b=eh80 conjugacy classes — view the full character table.
| maximal subgroup of | #72279905, #72280040, #72280067, … |
| maximal normal subgroup of | #72279905, #72280040, #72280067, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 34541);