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) = C22×C4 |
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 | C8×D8 |
| prime radical | C2×#135 | C2×C4 |
| LCS | G ▷ C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C8 ◁ C22×C16 ◁ C2×C4×C16 ◁ G | |
| LECS | G ▷ C2×C4×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
332 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)(5,6,10,18,13,20,11,17,14,21,9,19,12,30,29,36)(7,22,8,16,26,32,24,35,27,34,23,15,25,31,28,33);(2,4)(5,7,11,24,12,25,10,8,14,27,29,28,13,26,9,23)(6,15,19,32,20,33,36,34,21,16,18,31,30,35,17,22);(5,8,10,26,13,24,11,27,14,23,9,25,12,28,29,7)(6,16,18,32,20,35,17,34,21,15,19,31,30,33,36,22)Polycyclic presentation:
2^9|a^2=e,b^2=f,c^2=e,d^2=gi,e^2=h,f^2=g,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 | #72216211, #72216412, #72216414, … |
| maximal normal subgroup of | #72216211, #72216412, #72216414, … |
| maximal quotient of | #72225386 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 35870);