Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #7623323 |
| 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 | #60717 | C2 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C4×#41 ◁ C4×#131 ◁ G | |
| LECS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
1212 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)(19,25)(20,27)(21,28)(22,41)(23,45)(24,30)(26,37)(29,43)(31,47)(32,48)(33,63)(34,67)(35,50)(36,54)(38,56)(39,55)(40,57)(42,51)(44,70)(46,59)(49,65)(52,69)(53,68)(58,74)(60,76)(61,75)(62,77)(64,71)(66,80)(72,79)(73,78);(1,3,5,11)(2,6,8,9)(4,12,13,10)(7,15,16,14)(17,19,24,35)(18,25,30,50)(20,31,38,60)(21,32,23,34)(22,33,44,66)(26,46,53,73)(27,47,56,76)(28,48,45,67)(29,49,57,77)(36,58,39,61)(37,59,68,78)(40,62,43,65)(41,63,70,80)(42,64,69,79)(51,71,52,72)(54,74,55,75);(1,4,5,13)(2,7,8,16)(3,10,11,12)(6,14,9,15)(17,20)(18,26)(19,31)(21,36)(22,42)(23,39)(24,38)(25,46)(27,41)(28,51)(29,55)(30,53)(32,58)(33,64)(34,61)(35,60)(37,43)(40,68)(44,69)(45,52)(47,63)(48,71)(49,75)(50,73)(54,57)(56,70)(59,65)(62,78)(66,79)(67,72)(74,77)(76,80);(2,8)(6,9)(7,16)(14,15)(17,20)(18,27)(19,31)(21,39)(22,37)(23,36)(24,38)(25,47)(26,41)(28,55)(29,51)(30,56)(32,61)(33,59)(34,58)(35,60)(40,69)(42,43)(44,68)(45,54)(46,63)(48,75)(49,71)(50,76)(52,57)(53,70)(62,79)(64,65)(66,78)(67,74)(72,77)(73,80);(17,21,24,23)(18,28,30,45)(19,32,35,34)(20,36,38,39)(22,40,44,43)(25,48,50,67)(26,51,53,52)(27,54,56,55)(29,41,57,70)(31,58,60,61)(33,62,66,65)(37,42,68,69)(46,71,73,72)(47,74,76,75)(49,63,77,80)(59,64,78,79)Polycyclic presentation:
2^9|b^2=fi,c^2=f,e^2=i,g^2=hi,h^2=i,b^a=bf,c^a=cg,d^a=df,g^a=gh,h^a=hi,c^b=cf,g^c=gh,h^c=hi,e^d=ei80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7530633);