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 | #3531 | C4 |
| LCS | G ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C2×C8 ◁ G | |
| LECS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
1163 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)(17,18,21,27,22,28,37,50,23,25,38,45,39,46,61,69)(19,24,31,42,32,43,55,66,29,40,51,62,52,63,73,77)(20,26,34,48,35,49,58,72,36,44,59,67,60,68,76,79)(30,41,53,64,54,65,74,78,33,47,56,70,57,71,75,80);(1,3,5,11)(2,6,8,9)(4,12,13,10)(7,15,16,14)(17,19)(18,24)(20,33)(21,31)(22,32)(23,29)(25,40)(26,47)(27,42)(28,43)(30,36)(34,56)(35,57)(37,55)(38,51)(39,52)(41,44)(45,62)(46,63)(48,70)(49,71)(50,66)(53,59)(54,60)(58,75)(61,73)(64,67)(65,68)(69,77)(72,80)(74,76)(78,79);(1,4,5,13)(2,7,8,16)(3,10,11,12)(6,14,9,15)(18,25)(19,29)(26,44)(27,45)(28,46)(30,33)(31,51)(32,52)(48,67)(49,68)(50,69)(53,56)(54,57)(55,73)(72,79)(74,75);(2,8)(6,9)(7,16)(14,15)(19,29)(24,40)(30,33)(31,51)(32,52)(41,47)(42,62)(43,63)(53,56)(54,57)(55,73)(64,70)(65,71)(66,77)(74,75)(78,80);(17,20,21,34,22,35,37,58,23,36,38,59,39,60,61,76)(18,26,27,48,28,49,50,72,25,44,45,67,46,68,69,79)(19,30,31,53,32,54,55,74,29,33,51,56,52,57,73,75)(24,41,42,64,43,65,66,78,40,47,62,70,63,71,77,80)Polycyclic presentation:
2^9|a^2=g,b^2=f,c^2=f,e^2=g,g^2=h,h^2=i,b^a=bf,c^a=ci,d^a=df,c^b=cfi,d^b=di,e^b=ei152 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7529994);