Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #123011 |
| automorphism | ? |
| homology | Abelianization (H1) = C23 · Schur multiplier (H2) = C24 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #1595 |
| commutator, Frattini, cosocle | C2×C22⋊C8 | C23 |
| prime radical | #9974 | C2 |
| LCS | G ▷ C2×C22⋊C8 ▷ C22×C4 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C22×C4 ◁ C2×C22⋊C8 ◁ G | |
| LECS | G ▷ C2×C22⋊C8 ▷ C22×C4 ▷ C22 ▷ C1 | |
585 conjugacy classes of subgroups; only the 42 normal subgroups are shown.
This group has 585 conjugacy classes of subgroups — too many to draw. Only its 42 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 — 3 generators on 40 points:
(1,2)(3,5)(4,6)(7,8)(10,16)(11,20)(12,23)(13,14)(17,31)(18,35)(19,30)(21,37)(22,27)(24,28)(25,26)(29,32)(33,40)(34,38);(2,5)(6,8)(9,10,15,19)(11,17,22,34)(12,24,26,35)(13,18,14,28)(16,25,30,23)(20,32,27,33)(21,38,37,31)(29,39,40,36);(3,7)(9,11)(10,17)(12,21)(13,27)(14,20)(15,22)(16,29)(18,33)(19,34)(23,36)(24,38)(25,39)(26,37)(28,32)(30,40)(31,35)Polycyclic presentation:
2^9|b^2=h,d^2=fh,e^2=h,f^2=h,b^a=bd,c^a=ce,d^a=df,e^a=eh,f^a=fh,g^a=gi,d^b=df,e^b=egi,f^b=fh,d^c=dg,e^c=eh,e^d=ei41 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 30208);