Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #7076 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C24 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | #327 |
| commutator | C22×C8 | C22×C4 |
| Frattini, cosocle | C2×C22⋊C8 | C23 |
| socle | C22 | C2×#327 |
| prime radical | #23386 | C2 |
| LCS | G ▷ C22×C8 ▷ C22×C4 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C23×C4 ◁ C22×C22⋊C8 ◁ G | |
| LECS | G ▷ C2×C22⋊C8 ▷ C22×C4 ▷ C22 ▷ C1 | |
718 conjugacy classes of subgroups; only the 78 normal subgroups are shown.
This group has 718 conjugacy classes of subgroups — too many to draw. Only its 78 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 — 3 generators on 40 points:
(1,2)(3,7,8,5)(4,6)(9,10)(11,20,23,37)(12,17)(13,26,28,40)(14,30)(15,19)(16,22)(18,25,35,29)(21,36,24,34)(27,31)(32,38)(33,39);(2,5)(6,7)(9,11)(10,16)(12,21)(13,27)(14,24)(15,23)(17,31)(18,34)(19,33)(20,26)(22,29)(25,39)(28,38)(30,32)(35,36)(37,40);(9,12,15,14)(10,17,19,30)(11,21,23,24)(13,25,28,29)(16,31,33,32)(18,26,35,40)(20,36,37,34)(22,27,39,38)Polycyclic presentation:
2^9|a^2=e,c^2=i,d^2=gi,f^2=i,g^2=i,b^a=bd,d^a=df,f^a=fh,g^a=gi,d^b=dg,e^b=efgi,f^b=fi,g^b=gi,e^d=ehi68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 29736);