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)(17,18,21,26,22,27,37,48,23,28,38,49,39,50,61,72)(19,24,31,42,32,43,55,66,30,41,53,64,54,65,74,78)(20,25,35,46,36,47,60,71,34,45,58,69,59,70,76,80)(29,40,51,62,52,63,73,77,33,44,56,67,57,68,75,79);(1,3)(2,6)(4,11)(5,10)(7,9)(8,13)(12,16)(14,15)(17,19)(18,24)(20,33)(21,31)(22,32)(23,30)(25,44)(26,42)(27,43)(28,41)(29,34)(35,56)(36,57)(37,55)(38,53)(39,54)(40,45)(46,67)(47,68)(48,66)(49,64)(50,65)(51,58)(52,59)(60,75)(61,74)(62,69)(63,70)(71,79)(72,78)(73,76)(77,80);(3,10)(6,9)(11,16)(13,15)(17,20)(18,25)(19,29)(21,35)(22,36)(23,34)(24,40)(26,46)(27,47)(28,45)(30,33)(31,51)(32,52)(37,60)(38,58)(39,59)(41,44)(42,62)(43,63)(48,71)(49,69)(50,70)(53,56)(54,57)(55,73)(61,76)(64,67)(65,68)(66,77)(72,80)(74,75)(78,79);(2,7)(6,9)(8,14)(13,15)(20,34)(25,45)(29,33)(35,58)(36,59)(40,44)(46,69)(47,70)(51,56)(52,57)(60,76)(62,67)(63,68)(71,80)(73,75)(77,79);(1,4,5,12)(2,8,7,14)(3,11,10,16)(6,13,9,15)(19,30)(24,41)(29,33)(31,53)(32,54)(40,44)(42,64)(43,65)(51,56)(52,57)(55,74)(62,67)(63,68)(66,78)(73,75)(77,79)Polycyclic presentation:
2^9|a^2=g,e^2=f,g^2=h,h^2=i,b^a=bf,d^a=df,c^b=cfi,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, 7529947);