Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #509 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C4×C8 · Schur multiplier (H2) = C24 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C8 | C2×D8 |
| commutator | C8 | C2×C4×C8 |
| Frattini, cosocle | C2×C4×C8 | C23 |
| socle | C22 | C2×D4⋊C8 |
| prime radical | C24 | #104 |
| LCS | G ▷ C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C8 ◁ C23×C8 ◁ C22×C4×C8 ◁ G | |
| LECS | G ▷ C2×C4×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
556 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 3 generators on 36 points:
(1,2)(3,4)(5,6,9,14,11,17,26,31,12,15,25,30,10,16,8,24)(7,18,21,29,23,34,35,27,19,32,36,28,22,33,20,13);(2,4)(5,7,10,22,12,19,11,23)(6,13,15,27)(8,20,25,36,26,35,9,21)(14,18,30,32)(16,28,17,29)(24,33,31,34);(7,19)(13,27)(18,32)(20,35)(21,36)(22,23)(28,29)(33,34)Polycyclic presentation:
2^9|a^2=e,b^2=f,d^2=gi,e^2=h,f^2=gh,g^2=i,h^2=i,b^a=bd,d^a=dg,f^a=fgi,g^a=gi,c^b=ci128 conjugacy classes — view the full character table.
| maximal subgroup of | #72224803, #72224804, #72224809, … |
| maximal normal subgroup of | #72224803, #72224804, #72224809, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 33985);