Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #484587 |
| 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 | #18300 | 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 | |
1017 conjugacy classes of subgroups; only the 130 normal subgroups are shown.
This group has 1017 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)(11,19)(12,17)(13,15)(14,18)(16,22)(20,33)(21,35)(23,34)(24,31)(25,29)(26,28)(27,30)(32,37)(36,40)(38,39);(1,2)(3,6)(4,8)(5,7)(10,15)(12,23)(16,28)(17,32)(18,30)(19,35)(20,24)(25,37)(29,34)(31,40)(33,39)(36,38);(1,3,5,4)(2,6,7,8)(9,11)(10,16,18,19)(12,24)(13,26)(14,22)(15,28,30,35)(17,33,34,31)(20,23)(21,27)(25,36)(29,40,32,39)(37,38);(9,12,14,23)(10,17,18,34)(11,20,22,24)(13,25,27,37)(15,29,30,32)(16,31,19,33)(21,36,26,38)(28,39,35,40)Polycyclic presentation:
2^9|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,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, 391984);