Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #485617 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C25 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C2×B(4,𝔽2) |
| commutator, Frattini, cosocle | C2×C22⋊C4 | C24 |
| prime radical | #23443 | C2 |
| LCS | G ▷ C2×C22⋊C4 ▷ C23 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C22×C4 ◁ C2×#117 ◁ G | |
| LECS | G ▷ C2×C22⋊C4 ▷ C23 ▷ C2 ▷ C1 | |
1030 conjugacy classes of subgroups; only the 120 normal subgroups are shown.
This group has 1030 conjugacy classes of subgroups — too many to draw. Only its 120 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 40 points:
(2,5)(3,7)(6,8)(9,10)(11,21)(12,17)(13,18)(14,19)(15,20)(16,23)(22,35)(24,36)(25,40)(26,32)(27,33)(28,34)(29,37)(30,39)(31,38);(1,2)(3,8)(4,6)(5,7)(9,11,15,24)(10,16,20,30)(12,25,13,22)(14,23,28,39)(17,31,18,29)(19,21,34,36)(26,38,27,37)(32,40,33,35);(9,12,15,13)(10,17,20,18)(11,22,24,25)(14,27,28,26)(16,29,30,31)(19,32,34,33)(21,35,36,40)(23,38,39,37);(2,6)(5,8)(9,13,15,12)(10,18,20,17)(11,22,24,25)(14,27,28,26)(16,29,30,31)(19,33,34,32)(21,35,36,40)(23,37,39,38)Polycyclic presentation:
2^9|b^2=i,c^2=i,d^2=i,e^2=g,b^a=be,c^a=cf,e^a=eg,h^a=hi,c^b=ci,d^b=dg,e^b=eg,f^b=fhi,e^c=eh,f^e=fi44 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 392820);