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 |
| prime radical | #2186 | C22 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ #533 ◁ #2960 ◁ G | |
| LECS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
1203 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)(3,11)(4,8)(5,9)(6,10)(7,13)(12,16)(14,15)(17,18)(19,25)(20,34)(21,27)(22,40)(23,44)(24,29)(26,36)(28,42)(30,56)(31,47)(32,62)(33,50)(35,65)(37,66)(38,49)(39,55)(41,51)(43,69)(45,54)(46,58)(48,61)(52,70)(53,67)(57,74)(59,80)(60,77)(63,76)(64,73)(68,75)(71,78)(72,79);(1,3,6,13)(2,7,10,11)(4,12,5,14)(8,15,9,16)(17,19,24,33)(18,25,29,50)(20,30,37,60)(21,38,23,31)(22,32,43,64)(26,46,53,72)(27,49,44,47)(28,48,55,75)(34,56,66,77)(35,59,45,57)(36,58,67,79)(39,68,42,61)(40,62,69,73)(41,63,70,78)(51,76,52,71)(54,74,65,80);(1,4,6,5)(2,8,10,9)(3,12,13,14)(7,15,11,16)(17,20)(18,26)(19,30)(21,35)(22,41)(23,45)(24,37)(25,46)(27,51)(28,54)(29,53)(31,57)(32,63)(33,60)(34,40)(36,42)(38,59)(39,67)(43,70)(44,52)(47,71)(48,74)(49,76)(50,72)(55,65)(56,62)(58,61)(64,78)(66,69)(68,79)(73,77)(75,80);(1,5,6,4)(2,9,10,8)(3,12,13,14)(7,15,11,16);(2,10)(7,11)(8,9)(15,16)(17,21,24,23)(18,27,29,44)(19,31,33,38)(20,35,37,45)(22,39,43,42)(25,47,50,49)(26,51,53,52)(28,40,55,69)(30,57,60,59)(32,61,64,68)(34,65,66,54)(36,41,67,70)(46,71,72,76)(48,73,75,62)(56,74,77,80)(58,78,79,63)Polycyclic presentation:
2^9|b^2=fi,c^2=f,d^2=f,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.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7531665);