Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #501156 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C25×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C2×B(4,𝔽2) |
| commutator, Frattini, cosocle | C22×D4 | C24 |
| LCS | G ▷ C22×D4 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C22×C4 ◁ C2×C4×D4 ◁ G | |
| LECS | G ▷ C22×D4 ▷ C22 ▷ C2 ▷ C1 | |
1315 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)(6,10)(11,16)(12,17)(18,20);(1,2,4,3)(8,15)(13,19);(5,6,8,10)(7,11,13,16)(9,12,15,17)(14,18,19,20);(1,3,4,2)(5,7,9,14)(6,11,12,18)(8,13,15,19)(10,16,17,20)Polycyclic presentation:
2^9|b^2=e,c^2=f,d^2=ei,b^a=be,c^a=cf,d^a=de,g^a=ghi,c^b=cg,f^b=fh,g^c=gh,h^c=hi,g^f=gi53 conjugacy classes — view the full character table.
| maximal subgroup of | #72285915, #72286100, #72312510, … |
| maximal normal subgroup of | #72285915, #72286100, #72312510, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 410382);