Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #33605 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C28 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | C22×D8 |
| commutator, Frattini, cosocle | C2×C8 | C25 |
| socle | C22 | C23×D8 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C22×C42 ◁ C22×C4×C8 ◁ G | |
| LECS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
1418 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 40 points:
(1,2)(3,6)(4,5)(7,8)(9,10)(11,20)(12,17)(13,27)(14,29)(15,19)(16,24)(18,22)(21,33)(23,34)(25,39)(26,32)(28,40)(30,36)(31,38)(35,37);(2,5)(6,8)(11,21)(16,30)(20,33)(23,25)(24,36)(31,35)(34,39)(37,38);(9,11,22,37,14,25,28,36,15,21,26,38,12,23,13,24)(10,16,27,34,17,31,32,33,19,30,40,39,29,35,18,20);(1,3,4,7)(2,6,5,8)(11,21)(16,30)(20,33)(23,25)(24,36)(31,35)(34,39)(37,38);(9,12,15,14)(10,17,19,29)(11,23,21,25)(13,26,28,22)(16,31,30,35)(18,27,32,40)(20,34,33,39)(24,38,36,37)Polycyclic presentation:
2^9|c^2=gh,d^2=f,e^2=i,g^2=hi,h^2=i,b^a=bf,c^a=cg,g^a=gh,h^a=hi,c^b=ci,d^c=di104 conjugacy classes — view the full character table.
| maximal subgroup of | #72225180, #72225275, #72225279, … |
| maximal normal subgroup of | #72225180, #72225275, #72225279, … |
| maximal quotient of | #72343806 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7530444);