Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #7624842 |
| 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 | #121 | D8 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ #529 ◁ #2966 ◁ G | |
| LECS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
1133 conjugacy classes of subgroups.
| minimal degree | μ(G) = 80 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 80 points:
(1,2,5,8)(3,9,10,6)(4,12,13,7)(11,14,16,15)(17,18)(19,29)(20,34)(21,26)(22,41)(23,42)(24,31)(25,37)(27,35)(28,40)(30,52)(32,59)(33,45)(36,61)(38,68)(39,69)(43,71)(44,73)(46,66)(47,74)(48,65)(49,55)(50,53)(51,58)(54,60)(56,79)(57,70)(62,64)(63,67)(72,77)(75,80)(76,78);(1,3)(2,6)(4,11)(5,10)(7,14)(8,9)(12,15)(13,16)(17,19)(18,24)(20,30)(21,31)(22,32)(23,33)(25,43)(26,29)(27,44)(28,45)(34,60)(35,53)(36,54)(37,55)(38,56)(39,57)(40,58)(41,70)(42,51)(46,49)(47,72)(48,50)(52,61)(59,69)(62,75)(63,79)(64,77)(65,73)(66,71)(67,78)(68,76)(74,80);(17,20,35,64,23,38,39,66,21,36,65,74,40,67,22,37)(18,25,41,63,28,47,48,61,26,46,69,68,42,62,27,34)(19,30,53,77,33,56,57,71,31,54,73,80,58,78,32,55)(24,43,70,79,45,72,50,52,29,49,59,76,51,75,44,60);(3,10)(6,9)(11,16)(14,15)(19,31)(20,36)(24,29)(25,46)(32,57)(33,58)(34,61)(37,66)(38,67)(44,50)(45,51)(47,62)(53,73)(59,70)(63,68)(64,74);(1,4,5,13)(2,7,8,12)(3,11,10,16)(6,14,9,15)(18,26)(19,31)(25,46)(27,48)(28,42)(30,54)(32,57)(33,58)(34,61)(41,69)(47,62)(53,73)(55,71)(56,78)(63,68)(77,80)Polycyclic presentation:
2^9|a^2=fi,c^2=gh,e^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,d^c=di68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7532255);