Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| automorphism | ? |
| homology | Abelianization (H1) = C26 · Schur multiplier (H2) = C212 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, commutator, Frattini, socle, cosocle | C23 | C26 |
| prime radical | C24 | C25 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
3511 conjugacy classes of subgroups.
| minimal degree | μ(G) = 80 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 80 points:
(1,2)(3,9)(4,7)(5,8)(6,11)(10,15)(12,14)(13,16)(17,18)(19,20)(21,27)(22,35)(23,24)(25,26)(28,36)(29,31)(30,39)(32,41)(33,37)(34,46)(38,40)(42,45)(43,47)(44,48)(49,50,55,60)(51,56,61,67)(52,63,65,57)(53,58,66,73)(54,59,70,76)(62,71,69,75)(64,77,79,72)(68,74,80,78);(1,2,5,8)(3,6,11,9)(4,7,12,14)(10,13,16,15)(18,24)(19,26)(21,33)(28,35)(30,41)(31,38)(34,45)(44,47)(51,61)(53,66)(56,67)(58,73)(62,69)(68,80)(71,75)(74,78);(1,2)(3,6)(4,7)(5,8)(9,11)(10,13)(12,14)(15,16)(17,19)(18,25)(20,24)(21,29)(22,30)(23,26)(27,38)(28,39)(31,37)(32,35)(33,40)(34,43)(36,41)(42,47)(44,46)(45,48)(49,51)(50,56)(52,58)(53,57)(54,69)(55,61)(59,75)(60,67)(62,70)(63,66)(64,78)(65,73)(68,77)(71,76)(72,80)(74,79);(1,3,5,11)(2,6,8,9)(4,10,12,16)(7,13,14,15)(17,20,23,25)(18,26,24,19)(21,31,33,38)(22,32,36,39)(27,40,37,29)(28,41,35,30)(34,44,45,47)(42,48,46,43)(49,52,55,65)(50,57,60,63)(51,58,61,73)(53,67,66,56)(54,64,70,79)(59,72,76,77)(62,74,69,78)(68,75,80,71);(17,21,23,33)(18,27,24,37)(19,29,26,40)(20,31,25,38)(22,34,36,45)(28,42,35,46)(30,43,41,48)(32,44,39,47)(49,53)(50,58)(51,57)(52,56)(54,68)(55,66)(59,74)(60,73)(61,63)(62,72)(64,71)(65,67)(69,77)(70,80)(75,79)(76,78);(1,4,5,12)(2,7,8,14)(3,10,11,16)(6,13,9,15)(17,22,23,36)(18,28,24,35)(19,30,26,41)(20,32,25,39)(21,34,33,45)(27,42,37,46)(29,43,40,48)(31,44,38,47)(49,54)(50,59)(51,62)(52,64)(53,68)(55,70)(56,71)(57,72)(58,74)(60,76)(61,69)(63,77)(65,79)(66,80)(67,75)(73,78)Polycyclic presentation:
2^9|a^2=gi,b^2=gh,d^2=i,e^2=h,f^2=g,b^a=bg,c^a=ch,d^a=di,f^a=fh,c^b=ci,e^b=eghi,d^c=dh,f^c=fgi,e^d=egi74 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 9388711);