Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #135758 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C2×C8 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #1844 |
| commutator | C4×C8 | C22×C4 |
| Frattini, cosocle | C2×C4×C8 | C23 |
| prime radical | C23 | #492 |
| LCS | G ▷ C4×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C2×C42 ◁ C2×C82 ◁ G | |
| LECS | G ▷ C2×C4×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
302 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 3 generators on 24 points:
(2,5)(3,7)(6,8)(10,14,15,20)(11,17)(12,19)(16,23,22,21)(18,24);(9,10,11,14,12,16,17,21,13,15,18,20,19,22,24,23);(1,2,3,5,4,6,7,8)Polycyclic presentation:
2^9|a^2=f,b^2=d,c^2=e,d^2=g,e^2=h,g^2=i,b^a=bd,c^a=ce,d^a=dg,e^a=eh,g^a=gi,f^b=fi92 conjugacy classes — view the full character table.
| maximal subgroup of | #72213965, #72213983, #72214006, … |
| maximal normal subgroup of | #72213965, #72213983, #72214006, … |
| maximal quotient of | #72214006, #72216168, #72225164, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 42954);