Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #7624170 |
| 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 | #62267 | C2 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ #525 ◁ #2929 ◁ G | |
| LECS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
1213 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)(4,7)(5,8)(6,11)(10,15)(12,14)(13,16)(17,18,24,29)(19,25,31,47)(20,34,37,67)(21,27,23,44)(22,40,43,69)(26,51,54,36)(28,39,56,42)(30,57,58,78)(32,48,38,50)(33,64,62,75)(35,66,45,55)(41,52,70,53)(46,71,72,60)(49,68,74,63)(59,76,61,80)(65,77,79,73);(1,3)(2,6)(4,10)(5,11)(7,13)(8,9)(12,16)(14,15)(17,19)(18,25)(20,30)(21,38)(22,33)(23,32)(24,31)(26,46)(27,50)(28,49)(29,47)(34,57)(35,61)(36,60)(37,58)(39,68)(40,64)(41,65)(42,63)(43,62)(44,48)(45,59)(51,71)(52,77)(53,73)(54,72)(55,76)(56,74)(66,80)(67,78)(69,75)(70,79);(1,4,5,12)(2,7,8,14)(3,10,11,16)(6,13,9,15)(17,20)(18,26)(19,30)(21,35)(22,41)(23,45)(24,37)(25,46)(27,52)(28,55)(29,54)(31,58)(32,59)(33,65)(34,40)(36,42)(38,61)(39,51)(43,70)(44,53)(47,72)(48,73)(49,76)(50,77)(56,66)(57,64)(60,63)(62,79)(67,69)(68,71)(74,80)(75,78);(3,11)(6,9)(10,16)(13,15)(19,31)(25,47)(30,58)(32,38)(33,62)(46,72)(48,50)(49,74)(57,78)(59,61)(60,71)(63,68)(64,75)(65,79)(73,77)(76,80);(2,8)(6,9)(7,14)(13,15)(17,21,24,23)(18,27,29,44)(19,32,31,38)(20,35,37,45)(22,39,43,42)(25,48,47,50)(26,52,54,53)(28,40,56,69)(30,59,58,61)(33,63,62,68)(34,66,67,55)(36,41,51,70)(46,73,72,77)(49,75,74,64)(57,76,78,80)(60,79,71,65)Polycyclic presentation:
2^9|a^2=i,c^2=f,e^2=i,g^2=hi,h^2=i,b^a=bf,c^a=cg,e^a=ef,g^a=gh,h^a=hi,d^b=dfi,e^b=ei,g^c=gh,h^c=hi77 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7531508);