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 | #3486 | C22 |
| LCS | G ▷ C22×Q8 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C24 ◁ C23×Q8 ◁ G | |
| LECS | G ▷ C22×Q8 ▷ C22 ▷ C2 ▷ C1 | |
1004 conjugacy classes of subgroups; only the 140 normal subgroups are shown.
This group has 1004 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)(11,21)(12,18)(13,26)(14,29)(15,20)(16,23)(17,25)(19,31)(22,36)(24,39)(27,32)(28,35)(30,33)(34,40)(37,38);(2,5)(6,8)(11,22)(16,32)(21,36)(23,27)(24,30)(33,39)(34,37)(38,40);(1,3,4,7)(2,6,5,8)(9,11)(10,16)(12,23)(13,27)(14,30)(15,22)(17,33)(18,36)(19,34)(20,32)(21,26)(24,31)(25,40)(28,38)(29,37)(35,39);(9,12,15,13)(10,17,20,35)(11,23,22,27)(14,28,31,25)(16,33,32,39)(18,19,26,29)(21,37,36,34)(24,40,30,38)Polycyclic presentation:
2^9|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,c^b=ci,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, 419329);