Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #7624150 |
| 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 | #525 | D4 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ #525 ◁ #2929 ◁ G | |
| LECS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
1206 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)(19,25,33,50)(20,34,37,68)(21,27,23,44)(22,40,43,66)(26,51,54,36)(28,39,56,42)(30,57,61,78)(31,47,38,49)(32,63,65,74)(35,67,45,55)(41,52,70,53)(46,71,73,59)(48,69,76,62)(58,75,60,80)(64,77,79,72);(1,3)(2,6)(4,10)(5,11)(7,13)(8,9)(12,16)(14,15)(17,19)(18,25)(20,30)(21,38)(22,32)(23,31)(24,33)(26,46)(27,49)(28,48)(29,50)(34,57)(35,60)(36,59)(37,61)(39,69)(40,63)(41,64)(42,62)(43,65)(44,47)(45,58)(51,71)(52,77)(53,72)(54,73)(55,75)(56,76)(66,74)(67,80)(68,78)(70,79);(1,4,5,12)(2,7,8,14)(3,10,11,16)(6,13,9,15)(17,20,24,37)(18,26,29,54)(19,30,33,61)(21,35,23,45)(22,41,43,70)(25,46,50,73)(27,52,44,53)(28,55,56,67)(31,58,38,60)(32,64,65,79)(34,66,68,40)(36,39,51,42)(47,72,49,77)(48,75,76,80)(57,74,78,63)(59,69,71,62);(3,11)(6,9)(10,16)(13,15);(2,8)(6,9)(7,14)(13,15)(17,21,24,23)(18,27,29,44)(19,31,33,38)(20,35,37,45)(22,39,43,42)(25,47,50,49)(26,52,54,53)(28,40,56,66)(30,58,61,60)(32,62,65,69)(34,67,68,55)(36,41,51,70)(46,72,73,77)(48,74,76,63)(57,75,78,80)(59,79,71,64)Polycyclic presentation:
2^9|a^2=i,c^2=fi,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=df,e^b=ei,g^c=gh,h^c=hi71 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7531494);