Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #485337 |
| 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 | #3496 | C22 |
| LCS | G ▷ C2×C22⋊C4 ▷ C23 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C22×C4 ◁ C2×C4×D4 ◁ G | |
| LECS | G ▷ C2×C22⋊C4 ▷ C23 ▷ C2 ▷ C1 | |
828 conjugacy classes of subgroups; only the 118 normal subgroups are shown.
This group has 828 conjugacy classes of subgroups — too many to draw. Only its 118 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) = 128 |
A minimal faithful permutation representation — 4 generators on 40 points:
(2,6)(3,4)(7,8)(9,10)(11,19)(12,23)(13,17)(14,18)(15,21)(16,24)(20,34)(22,33)(25,40)(26,30)(27,31)(28,37)(29,38)(32,35)(36,39);(1,2)(3,6)(4,8)(5,7)(9,11,14,22)(10,15,18,29)(12,20,24,35)(13,21,27,38)(16,28,23,39)(17,19,31,33)(25,36,26,37)(30,32,40,34);(9,12,14,24)(10,16)(11,20,22,35)(13,26,27,25)(15,28)(17,30)(18,23)(19,32)(21,37,38,36)(29,39)(31,40)(33,34);(1,3,5,4)(2,6,7,8)(12,24)(16,23)(20,35)(25,26)(28,39)(30,40)(32,34)(36,37)Polycyclic presentation:
2^9|b^2=i,c^2=h,d^2=g,e^2=g,b^a=be,c^a=cf,d^a=dg,e^a=eg,h^a=hi,e^b=eg,f^b=fhi,d^c=di,e^c=eh,f^c=fi,f^e=fi41 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 392684);