Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #125644 |
| automorphism | ? |
| homology | Abelianization (H1) = C23 · Schur multiplier (H2) = C22 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #1595 |
| commutator, Frattini, cosocle | C2×#97 | C23 |
| prime radical | #10032 | C2 |
| LCS | G ▷ C2×#97 ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C22×C4 ◁ C2×#97 ◁ G | |
| LECS | G ▷ C2×#97 ▷ C22×C4 ▷ C2 ▷ C1 | |
349 conjugacy classes of subgroups; only the 52 normal subgroups are shown.
This group has 349 conjugacy classes of subgroups — too many to draw. Only its 52 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:
(2,6)(3,4)(7,8)(9,10)(11,22)(12,21)(13,17)(14,20)(15,18)(16,23)(19,28)(24,36)(25,35)(26,32)(27,33)(29,34)(30,39)(31,40)(37,38);(1,2)(3,6)(4,8)(5,7)(9,11)(10,16)(12,26)(13,24)(14,25)(15,23)(17,30)(18,35)(19,36)(20,32)(21,22)(27,39)(28,38)(29,40)(31,33)(34,37);(1,3,5,4)(2,6,7,8)(10,17)(12,15)(13,28)(16,30)(18,33)(19,21)(20,34)(22,36)(23,26)(24,38)(27,29)(31,35)(32,37)(39,40)Polycyclic presentation:
2^9|c^2=fi,d^2=f,e^2=hi,h^2=i,b^a=bd,c^a=ce,d^a=df,e^a=eh,g^a=gi,h^a=hi,d^b=df,e^b=egi,d^c=dg,e^c=eh,h^c=hi,e^d=ei41 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 32887);