Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #484583 |
| 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 | |
1031 conjugacy classes of subgroups; only the 130 normal subgroups are shown.
This group has 1031 conjugacy classes of subgroups — too many to draw. Only its 130 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:
(2,6)(3,4)(7,8)(9,10)(11,21)(12,26)(13,18)(14,19)(15,20)(16,24)(17,28)(22,32)(23,37)(25,38)(27,36)(29,31)(30,34)(33,39)(35,40);(1,2)(3,6)(4,8)(5,7)(9,11,15,25)(10,16,20,33)(12,22,28,40)(13,29,27,23)(14,24,30,39)(17,31,26,37)(18,35,36,32)(19,21,34,38);(1,3,5,4)(2,6,7,8)(9,12)(10,17,20,26)(11,22)(13,14)(15,28)(16,31,33,37)(18,34,36,19)(21,35,38,32)(23,39)(24,29)(25,40)(27,30);(9,13,15,27)(10,18,20,36)(11,23,25,29)(12,14,28,30)(16,32,33,35)(17,34,26,19)(21,37,38,31)(22,39,40,24)Polycyclic presentation:
2^9|b^2=i,c^2=ghi,d^2=i,e^2=g,b^a=be,c^a=cf,e^a=eg,h^a=hi,d^b=di,e^b=eg,f^b=fhi,e^c=eh,f^c=fi,f^e=fi50 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 391973);