Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #346950 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C28 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #2846 |
| commutator, Frattini, cosocle | C2×C42 | C24 |
| LCS | G ▷ C2×C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C2×C42 ◁ G | |
| LECS | G ▷ C2×C42 ▷ C22 ▷ C1 | |
3170 conjugacy classes of subgroups; only the 222 normal subgroups are shown.
This group has 3170 conjugacy classes of subgroups — too many to draw. Only its 222 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)(3,5)(4,7)(6,8)(11,19)(13,24)(15,27)(16,30)(17,25)(20,33)(21,26)(23,35)(28,39)(29,32)(34,37)(38,40);(2,5)(3,8)(6,7)(9,10)(11,15)(12,16)(13,25)(14,18)(17,24)(19,32)(20,28)(21,29)(22,30)(23,36)(26,27)(31,35)(33,39)(34,40)(37,38);(9,11,14,21)(10,15,18,29)(12,20,22,34)(13,26,24,19)(16,28,30,40)(17,32,25,27)(23,37,35,33)(31,39,36,38);(9,12)(10,16)(11,20)(13,23)(14,22)(15,28)(17,31)(18,30)(19,33)(21,34)(24,35)(25,36)(26,37)(27,38)(29,40)(32,39)Polycyclic presentation:
2^9|c^2=i,e^2=h,f^2=i,b^a=be,c^a=cf,d^a=dg,e^a=eh,f^a=fi,e^b=eh,f^b=fi,g^b=gi,e^c=ei,f^c=fi,e^d=ei62 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 254152);