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)(17,18,21,27,22,28,37,49,23,25,38,44,39,45,61,68)(19,24,31,42,32,43,56,67,29,40,51,62,52,63,72,77)(20,33,34,57,35,58,59,76,36,26,46,47,60,48,69,70)(30,53,54,73,55,74,75,80,50,41,64,65,71,66,78,79);(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)(25,40)(26,41)(27,42)(28,43)(33,53)(34,54)(35,55)(36,50)(37,56)(38,51)(39,52)(44,62)(45,63)(46,64)(47,65)(48,66)(49,67)(57,73)(58,74)(59,75)(60,71)(61,72)(68,77)(69,78)(70,79)(76,80);(1,4,5,13)(2,7,8,16)(3,10,11,12)(6,14,9,15)(18,25)(19,29)(26,33)(27,44)(28,45)(30,50)(31,51)(32,52)(47,57)(48,58)(49,68)(54,64)(55,71)(56,72)(70,76)(75,78);(2,8)(6,9)(7,16)(14,15)(19,29)(24,40)(30,50)(31,51)(32,52)(41,53)(42,62)(43,63)(54,64)(55,71)(56,72)(65,73)(66,74)(67,77)(75,78)(79,80);(17,20,21,34,22,35,37,59,23,36,38,46,39,60,61,69)(18,26,27,47,28,48,49,70,25,33,44,57,45,58,68,76)(19,30,31,54,32,55,56,75,29,50,51,64,52,71,72,78)(24,41,42,65,43,66,67,79,40,53,62,73,63,74,77,80)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,e^a=ei,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, 7529988);