Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #484162 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C26 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C2×B(4,𝔽2) |
| commutator, Frattini, cosocle | C2×C22⋊C4 | C24 |
| LCS | G ▷ C2×C22⋊C4 ▷ C23 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C22×C4 ◁ C2×#117 ◁ G | |
| LECS | G ▷ C2×C22⋊C4 ▷ C23 ▷ C2 ▷ C1 | |
1240 conjugacy classes of subgroups; only the 138 normal subgroups are shown.
This group has 1240 conjugacy classes of subgroups — too many to draw. Only its 138 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:
(1,2)(3,5)(4,7)(6,8)(9,10)(11,19)(12,16)(13,17)(14,18)(15,21)(20,31)(22,32)(23,28)(24,29)(25,30)(26,34)(27,35)(33,39)(36,38)(37,40);(2,5)(3,8)(6,7)(9,11,14,22)(10,15,18,27)(12,20,23,33)(13,21,25,35)(16,26,28,37)(17,19,30,32)(24,34,36,40)(29,31,38,39);(12,23)(13,25)(16,28)(20,33)(21,35)(26,37)(29,38)(31,39);(9,12,14,23)(10,16,18,28)(11,20,22,33)(13,24,25,36)(15,26,27,37)(17,29,30,38)(19,31,32,39)(21,34,35,40)Polycyclic presentation:
2^9|b^2=i,d^2=i,e^2=g,b^a=be,c^a=cf,e^a=eg,h^a=hi,e^b=eg,f^b=fhi,d^c=di,e^c=eh,f^e=fi50 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 391361);