Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #11677 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C42 · Schur multiplier (H2) = C23 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | C2×#102 |
| commutator | C2×C8 | C2×C42 |
| Frattini, cosocle | C2×C4×C8 | C23 |
| socle | C22 | C2×#340 |
| prime radical | C24 | #103 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C22×C42 ◁ C22×C4×C8 ◁ G | |
| LECS | G ▷ C2×C4×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
459 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 3 generators on 40 points:
(1,2,4,8,5,7,3,6)(9,10,12,20,16,22,15,18)(11,23,24,35,28,38,27,39)(13,30,29,37,32,21,14,19)(17,26,34,31,36,40,33,25);(2,6,7,8)(9,11,14,26,15,27,32,25,16,28,29,40,12,24,13,31)(10,17,21,35,20,34,19,38,22,36,30,39,18,33,37,23);(2,7)(6,8)(9,12,16,15)(10,18,22,20)(11,24,28,27)(13,29,32,14)(17,33,36,34)(19,21,37,30)(23,39,38,35)(25,26,31,40)Polycyclic presentation:
2^9|a^2=e,b^2=f,c^2=i,d^2=gi,e^2=h,f^2=ghi,g^2=i,b^a=bd,c^a=ch,d^a=dg,f^a=fg,g^a=gi,d^b=di104 conjugacy classes — view the full character table.
| maximal subgroup of | #72285908, #72285969, #72286002 |
| maximal normal subgroup of | #72285908, #72285969, #72286002 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 34061);