Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #7624170 |
| 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 ◁ #525 ◁ #2929 ◁ G | |
| LECS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
1550 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)(3,7)(4,6)(5,8)(11,19)(12,22)(13,24)(15,27)(17,32)(18,31)(20,37)(21,36)(23,26)(25,35)(28,39)(29,40)(30,38)(33,34);(1,3)(2,5)(4,8)(6,7)(9,10,14,16)(11,15,21,28)(12,17,25,30)(13,18,24,31)(19,27,36,39)(20,29,26,34)(22,32,35,38)(23,33,37,40);(9,11,14,21)(10,15,16,28)(12,23,25,37)(13,26,24,20)(17,33,30,40)(18,34,31,29)(19,35,36,22)(27,38,39,32);(3,8)(5,7)(10,16)(15,28)(17,30)(18,31)(27,39)(29,34)(32,38)(33,40);(2,6)(5,7)(10,16)(15,28)(17,30)(18,31)(27,39)(29,34)(32,38)(33,40)Polycyclic presentation:
2^9|b^2=i,c^2=i,g^2=hi,h^2=i,b^a=bf,c^a=cg,e^a=ef,g^a=gh,h^a=hi,d^b=dfi,e^b=ei,g^c=gh,h^c=hi77 conjugacy classes — view the full character table.
| maximal subgroup of | #72314433, #72315814, #72315829, … |
| maximal normal subgroup of | #72314433, #72315814, #72315829, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7531370);