Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #7624150 |
| 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 | |
1555 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)(10,16)(11,20)(12,15)(14,26)(17,31)(19,36)(21,39)(22,38)(24,25)(27,28)(29,33)(30,35)(32,40)(34,37);(1,3)(2,5)(4,8)(6,7)(9,10)(11,17)(12,23)(13,16)(14,19)(15,18)(20,33)(21,34)(22,29)(24,36)(25,30)(26,35)(27,37)(28,32)(31,38)(39,40);(9,11,13,22)(10,17,16,29)(12,21,15,28)(14,27,25,39)(18,32,23,34)(19,37,30,40)(20,24,38,26)(31,35,33,36);(3,8)(5,7)(10,16)(17,29)(18,23)(19,30)(31,33)(32,34)(35,36)(37,40);(2,6)(5,7)(9,12,13,15)(10,18,16,23)(11,21,22,28)(14,24,25,26)(17,32,29,34)(19,35,30,36)(20,27,38,39)(31,37,33,40)Polycyclic presentation:
2^9|c^2=i,e^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=df,e^b=ei,g^c=gh,h^c=hi71 conjugacy classes — view the full character table.
| maximal subgroup of | #72314433, #72315814, #72315824, … |
| maximal normal subgroup of | #72314433, #72315814, #72315824, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7531358);