Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #7624888 |
| 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 |
| prime radical | #33823 | C2 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ #533 ◁ #2960 ◁ G | |
| LECS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
1156 conjugacy classes of subgroups.
| minimal degree | μ(G) = 80 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 80 points:
(1,2)(3,9)(4,7)(5,8)(6,11)(10,15)(12,14)(13,16)(17,18)(19,29)(20,34)(21,26)(22,41)(23,43)(24,31)(25,37)(27,42)(28,40)(30,54)(32,61)(33,46)(35,50)(36,64)(38,71)(39,47)(44,73)(45,55)(48,68)(49,67)(51,57)(52,62)(53,60)(56,63)(58,80)(59,72)(65,70)(66,69)(74,79)(75,77)(76,78);(1,3,5,11)(2,6,8,9)(4,12,13,10)(7,15,16,14)(17,19,21,31)(18,24,26,29)(20,30,36,56)(22,32,39,59)(23,33,40,60)(25,44,48,51)(27,45,50,52)(28,46,43,53)(34,63,64,54)(35,55,42,62)(37,57,68,73)(38,58,69,78)(41,72,47,61)(49,74,65,75)(66,80,71,76)(67,77,70,79);(17,20,35,67,23,38,22,37,21,36,42,70,40,69,39,68)(18,25,47,71,28,49,27,34,26,48,41,66,43,65,50,64)(19,30,55,77,33,58,32,57,31,56,62,79,60,78,59,73)(24,44,61,76,46,74,45,63,29,51,72,80,53,75,52,54);(1,4,5,13)(2,7,8,16)(3,10,11,12)(6,14,9,15)(19,31)(24,29)(30,56)(32,59)(33,60)(44,51)(45,52)(46,53)(54,63)(55,62)(57,73)(58,78)(61,72)(74,75)(76,80)(77,79);(2,8)(6,9)(7,16)(14,15)(18,26)(19,31)(20,36)(27,50)(28,43)(32,59)(33,60)(37,68)(38,69)(41,47)(44,51)(54,63)(55,62)(67,70)(74,75)(76,80)Polycyclic presentation:
2^9|b^2=f,c^2=ghi,d^2=fi,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,e^c=ei74 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7532166);