Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #511785 |
| 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, prime radical, cosocle | C22×D4 | C24 |
| LCS | G ▷ C22×D4 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C22×C4 ◁ C2×C4×D4 ◁ G | |
| LECS | G ▷ C22×D4 ▷ C22 ▷ C2 ▷ C1 | |
911 conjugacy classes of subgroups; only the 152 normal subgroups are shown.
This group has 911 conjugacy classes of subgroups — too many to draw. Only its 152 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:
(1,2,4,6)(3,7,8,5)(9,10)(11,19)(12,16)(13,17)(14,18)(15,22)(20,34)(21,35)(23,30)(24,39)(25,32)(26,40)(27,33)(28,36)(29,37)(31,38);(1,3,4,8)(2,5,6,7)(11,20)(12,23)(15,28)(16,30)(19,34)(22,36)(25,26)(32,40);(9,11,14,20)(10,15,18,28)(12,24,23,21)(13,22,27,36)(16,31,30,29)(17,19,33,34)(25,38,26,37)(32,39,40,35);(9,12,14,23)(10,16,18,30)(11,21,20,24)(13,26,27,25)(15,29,28,31)(17,32,33,40)(19,35,34,39)(22,38,36,37)Polycyclic presentation:
2^9|a^2=e,b^2=e,c^2=i,d^2=i,b^a=be,c^a=cf,d^a=dg,h^a=hi,c^b=ci,d^b=di,d^c=di,g^c=ghi,f^d=fh,g^f=gi53 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 419215);