Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #3532 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C8 · Schur multiplier (H2) = C28 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, Frattini, cosocle | C2×C8 | C25 |
| commutator, socle | C22 | C24×C8 |
| prime radical | #3455 | C4 |
| LCS | G ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C2×C8 ◁ G | |
| LECS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
1225 conjugacy classes of subgroups.
| minimal degree | μ(G) = 80 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 80 points:
(1,2)(3,9)(4,8)(5,7)(6,10)(11,15)(12,14)(13,16)(17,18,21,26,22,27,36,46,23,28,37,47,38,48,58,68)(19,24,31,41,32,42,54,64,30,40,52,62,53,63,72,77)(20,25,34,44,35,45,57,67,33,43,55,65,56,66,73,78)(29,39,50,60,51,61,71,76,49,59,69,74,70,75,79,80);(1,3)(2,6)(4,11)(5,10)(7,9)(8,13)(12,16)(14,15)(17,19)(18,24)(20,29)(21,31)(22,32)(23,30)(25,39)(26,41)(27,42)(28,40)(33,49)(34,50)(35,51)(36,54)(37,52)(38,53)(43,59)(44,60)(45,61)(46,64)(47,62)(48,63)(55,69)(56,70)(57,71)(58,72)(65,74)(66,75)(67,76)(68,77)(73,79)(78,80);(3,10)(6,9)(11,16)(13,15)(17,20)(18,25)(19,29)(21,34)(22,35)(23,33)(24,39)(26,44)(27,45)(28,43)(30,49)(31,50)(32,51)(36,57)(37,55)(38,56)(40,59)(41,60)(42,61)(46,67)(47,65)(48,66)(52,69)(53,70)(54,71)(58,73)(62,74)(63,75)(64,76)(68,78)(72,79)(77,80);(2,7)(6,9)(8,14)(13,15)(20,33)(25,43)(29,49)(34,55)(35,56)(39,59)(44,65)(45,66)(50,69)(51,70)(57,73)(60,74)(61,75)(67,78)(71,79)(76,80);(1,4,5,12)(2,8,7,14)(3,11,10,16)(6,13,9,15)(19,30)(24,40)(29,49)(31,52)(32,53)(39,59)(41,62)(42,63)(50,69)(51,70)(54,72)(60,74)(61,75)(64,77)(71,79)(76,80)Polycyclic presentation:
2^9|a^2=g,e^2=f,g^2=h,h^2=i,b^a=bf,d^a=df,c^b=cf,e^b=ei,d^c=di152 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7529946);