Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #3264 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C8 · Schur multiplier (H2) = C24 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C8 | C22×D4 |
| commutator | C2×C4 | C23×C8 |
| Frattini, cosocle | C4×C8 | C24 |
| socle | C22 | C2×C8×D4 |
| prime radical | #447 | C8 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C8 ◁ C4×M32 ◁ G | |
| LECS | G ▷ C4×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
465 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 4 generators on 40 points:
(1,2)(3,5)(4,6)(7,8)(9,10,12,16,13,17,23,29,14,18,24,30,25,31,36,39)(11,19,21,32,22,33,35,40,20,15,26,27,34,28,37,38);(2,5)(3,8)(6,7);(9,11,12,21,13,22,23,35,14,20,24,26,25,34,36,37)(10,15,16,27,17,28,29,38,18,19,30,32,31,33,39,40);(2,6)(5,7)(11,20)(15,19)(21,26)(22,34)(27,32)(28,33)(35,37)(38,40)Polycyclic presentation:
2^9|a^2=f,c^2=f,e^2=g,f^2=h,h^2=i,b^a=be,c^a=ci,d^a=dg,e^a=eg,e^b=eg,d^c=di116 conjugacy classes — view the full character table.
| maximal subgroup of | #72320272, #72322022, #72322062 |
| maximal normal subgroup of | #72320272, #72322022, #72322062 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 390834);