Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #7624184 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C28 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C23×D8 |
| commutator, Frattini, cosocle | C2×C8 | C25 |
| prime radical | #62325 | C2 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ #532 ◁ #2931 ◁ G | |
| LECS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
1206 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)(17,18,23,29)(19,24,30,43)(20,26,34,50)(21,25,39,49)(22,40,38,28)(27,53,48,37)(31,42,54,69)(32,46,55,41)(33,61,56,45)(35,47,62,72)(36,65,63,52)(44,59,70,60)(51,74,73,64)(57,71,75,66)(58,79,76,68)(67,77,80,78);(1,3)(2,6)(4,10)(5,11)(7,13)(8,9)(12,16)(14,15)(17,19)(18,24)(20,31)(21,32)(22,33)(23,30)(25,46)(26,42)(27,44)(28,45)(29,43)(34,54)(35,57)(36,58)(37,60)(38,56)(39,55)(40,61)(41,49)(47,71)(48,70)(50,69)(51,67)(52,68)(53,59)(62,75)(63,76)(64,78)(65,79)(66,72)(73,80)(74,77);(1,4,5,12)(2,7,8,14)(3,10,11,16)(6,13,9,15)(18,25)(19,30)(21,37)(22,38)(24,41)(26,47)(27,28)(29,49)(31,54)(32,59)(35,64)(36,63)(39,53)(40,48)(42,66)(43,46)(44,61)(45,70)(50,72)(51,52)(55,60)(57,77)(62,74)(65,73)(67,79)(68,80)(69,71)(75,78);(3,11)(6,9)(10,16)(13,15)(17,20)(18,26)(19,31)(21,35)(22,36)(23,34)(24,42)(25,47)(27,51)(28,52)(29,50)(30,54)(32,57)(33,58)(37,64)(38,63)(39,62)(40,65)(41,66)(43,69)(44,67)(45,68)(46,71)(48,73)(49,72)(53,74)(55,75)(56,76)(59,77)(60,78)(61,79)(70,80);(2,8)(6,9)(7,14)(13,15)(19,30)(20,34)(24,43)(26,50)(32,55)(33,56)(35,62)(36,63)(41,46)(44,70)(45,61)(47,72)(51,73)(52,65)(59,60)(64,74)Polycyclic presentation:
2^9|a^2=i,c^2=f,g^2=hi,h^2=i,b^a=bf,c^a=cg,e^a=ef,g^a=gh,h^a=hi,c^b=ci,d^b=df,e^b=ei,g^c=gh,h^c=hi,e^d=ei65 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7531531);