Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #7623626 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C28 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C23×D8 |
| commutator, Frattini, cosocle | C2×C8 | C25 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ #529 ◁ #2966 ◁ G | |
| LECS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
1325 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 40 points:
(2,5)(6,8)(9,10)(11,15)(12,16)(13,26)(14,17)(18,23)(19,24)(20,30)(21,27)(22,36)(25,31)(28,35)(29,34)(32,37)(33,39)(38,40);(1,2)(3,6)(4,5)(7,8)(9,11,14,20)(10,15,17,30)(12,21,25,33)(13,22,24,34)(16,27,31,39)(18,28,32,38)(19,29,26,36)(23,35,37,40);(10,16)(12,23)(13,24)(15,27)(17,31)(18,19)(21,35)(22,34)(25,37)(26,32)(28,29)(30,39)(33,40)(36,38);(2,5)(6,8);(1,3,4,7)(2,6,5,8)(10,17)(11,20)(16,31)(18,32)(19,26)(21,33)(22,34)(35,40)Polycyclic presentation:
2^9|b^2=i,e^2=f,g^2=hi,h^2=i,b^a=bf,c^a=cg,e^a=ei,g^a=gh,h^a=hi,d^b=df,e^b=ei,g^c=gh,h^c=hi80 conjugacy classes — view the full character table.
| maximal subgroup of | #72216687, C2×#7624105, #72320093, … |
| maximal normal subgroup of | #72216687, C2×#7624105, #72320093, … |
| maximal quotient of | C2×#7624105, #72319537, #72319542, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7531301);