Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #560 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C8 · Schur multiplier (H2) = C28 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C16 | C24 |
| commutator, socle | C22 | C24×C8 |
| Frattini, cosocle | C2×C8 | C25 |
| prime radical | #3531 | C4 |
| LCS | G ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C2×C16 ◁ 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)(17,18,21,27,22,28,36,48,23,25,37,44,38,45,58,67)(19,24,31,41,32,42,54,64,29,39,50,60,51,61,71,76)(20,26,33,46,34,47,55,68,35,43,56,65,57,66,73,78)(30,40,52,62,53,63,72,77,49,59,69,74,70,75,79,80);(1,3,5,11)(2,6,8,9)(4,12,13,10)(7,15,16,14)(17,19)(18,24)(20,30)(21,31)(22,32)(23,29)(25,39)(26,40)(27,41)(28,42)(33,52)(34,53)(35,49)(36,54)(37,50)(38,51)(43,59)(44,60)(45,61)(46,62)(47,63)(48,64)(55,72)(56,69)(57,70)(58,71)(65,74)(66,75)(67,76)(68,77)(73,79)(78,80);(1,4,5,13)(2,7,8,16)(3,10,11,12)(6,14,9,15)(18,25)(19,29)(26,43)(27,44)(28,45)(30,49)(31,50)(32,51)(46,65)(47,66)(48,67)(52,69)(53,70)(54,71)(68,78)(72,79);(2,8)(6,9)(7,16)(14,15)(19,29)(24,39)(30,49)(31,50)(32,51)(40,59)(41,60)(42,61)(52,69)(53,70)(54,71)(62,74)(63,75)(64,76)(72,79)(77,80);(17,20,21,33,22,34,36,55,23,35,37,56,38,57,58,73)(18,26,27,46,28,47,48,68,25,43,44,65,45,66,67,78)(19,30,31,52,32,53,54,72,29,49,50,69,51,70,71,79)(24,40,41,62,42,63,64,77,39,59,60,74,61,75,76,80)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=di152 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7529980);