Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #484181 |
| 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×C4⋊C4 | C24 |
| LCS | G ▷ C2×C4⋊C4 ▷ C23 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C22×C4 ◁ C2×#117 ◁ G | |
| LECS | G ▷ C2×C4⋊C4 ▷ C23 ▷ C2 ▷ C1 | |
1196 conjugacy classes of subgroups; only the 138 normal subgroups are shown.
This group has 1196 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)(11,15)(12,17)(13,16)(14,19)(18,26)(20,28)(21,27)(22,30)(23,33)(24,31)(25,29)(32,36)(34,40)(35,38)(37,39);(2,5)(3,8)(6,7)(9,11,14,22)(10,15,19,30)(12,20,23,34)(13,25,26,21)(16,27,18,29)(17,28,33,40)(24,37,36,35)(31,38,32,39);(10,16)(12,23)(13,26)(15,27)(17,32)(18,19)(20,34)(21,25)(28,39)(29,30)(31,33)(38,40);(9,12,14,23)(10,17,19,33)(11,20,22,34)(13,24,26,36)(15,28,30,40)(16,31,18,32)(21,35,25,37)(27,38,29,39)Polycyclic presentation:
2^9|b^2=i,d^2=i,e^2=g,f^2=i,b^a=be,c^a=cf,e^a=eg,f^a=fi,h^a=hi,e^b=eg,f^b=fhi,d^c=di,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, 391380);