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 | #33744 | 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)(17,18,21,26)(19,29,31,24)(20,34,36,63)(22,41,39,69)(23,42,40,28)(25,46,47,37)(27,50,49,35)(30,54,56,62)(32,61,59,70)(33,45,60,53)(38,68,67,65)(43,57,51,72)(44,55,52,74)(48,66,64,73)(58,79,78,76)(71,80,75,77);(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)(25,43,47,51)(27,44,49,52)(28,45,42,53)(34,62,63,54)(35,55,50,74)(37,57,46,72)(38,58,67,78)(41,70,69,61)(48,71,64,75)(65,79,68,76)(66,77,73,80);(17,20,35,66,23,38,39,46,21,36,50,73,40,67,22,37)(18,25,41,65,28,48,49,63,26,47,69,68,42,64,27,34)(19,30,55,77,33,58,59,72,31,56,74,80,60,78,32,57)(24,43,70,79,45,71,52,54,29,51,61,76,53,75,44,62);(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)(43,51)(44,52)(45,53)(54,62)(55,74)(57,72)(58,78)(61,70)(71,75)(76,79)(77,80);(2,8)(6,9)(7,16)(14,15)(18,26)(19,31)(20,36)(27,49)(28,42)(32,59)(33,60)(37,46)(38,67)(41,69)(43,51)(54,62)(55,74)(66,73)(71,75)(76,79)Polycyclic presentation:
2^9|a^2=i,b^2=f,c^2=gh,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, 7532165);