Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #511802 |
| 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 | C22×Q8 | C24 |
| prime radical | #3487 | C22 |
| LCS | G ▷ C22×Q8 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C22×C4 ◁ C22×#41 ◁ G | |
| LECS | G ▷ C22×Q8 ▷ C22 ▷ C2 ▷ C1 | |
1019 conjugacy classes of subgroups; only the 140 normal subgroups are shown.
This group has 1019 conjugacy classes of subgroups — too many to draw. Only its 140 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)(3,6)(4,5)(7,8)(9,10)(11,23)(12,17)(13,21)(14,20)(15,30)(16,32)(18,26)(19,28)(22,29)(24,34)(25,35)(27,38)(31,37)(33,40)(36,39);(2,5)(6,8)(9,11,14,12)(10,17,20,23)(13,26,15,24)(16,25,29,27)(18,21,34,30)(19,36,37,33)(22,35,32,38)(28,40,31,39);(1,3,4,7)(2,6,5,8)(9,12,14,11)(10,18,20,34)(13,26,15,24)(16,25,29,27)(17,21,23,30)(19,35,37,38)(22,36,32,33)(28,39,31,40);(9,13,14,15)(10,19,20,37)(11,24,12,26)(16,31,29,28)(17,33,23,36)(18,35,34,38)(21,22,30,32)(25,40,27,39)Polycyclic presentation:
2^9|b^2=i,c^2=e,d^2=i,f^2=i,g^2=i,b^a=be,c^a=cf,d^a=dg,f^a=fi,g^a=gi,h^a=hi,d^b=di,f^c=fi,g^c=ghi,f^d=fh,g^d=gi,g^f=gi53 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 419337);