Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #395173 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C22×C42 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | #535 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| socle | C22 | #2598 |
| prime radical | C2×#575 | C22 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
998 conjugacy classes of subgroups.
| minimal degree | μ(G) = 20 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 20 points:
(2,3)(5,6,8,12,10,14,9,13)(7,15,16,20,17,11,18,19);(1,2,4,3)(5,7,9,16,10,17,8,18)(6,11,13,19,14,15,12,20);(1,3,4,2)(6,12,14,13)(11,19,15,20);(6,13,14,12)(11,19,15,20)Polycyclic presentation:
2^9|a^2=ef,b^2=eh,c^2=e,d^2=g,f^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,f^a=fi,g^a=gi,h^a=hi,c^b=cg,f^b=fi80 conjugacy classes — view the full character table.
| maximal subgroup of | #72320063 |
| maximal normal subgroup of | #72320063 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 318712);