Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #7623308 |
| 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 | #33770 | C2 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C4×#41 ◁ C4×#131 ◁ G | |
| LECS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
1237 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)(4,8)(5,7)(6,10)(11,15)(12,14)(13,16)(17,18)(19,24)(20,33)(21,37)(22,42)(23,26)(25,34)(27,40)(28,39)(29,53)(30,45)(31,59)(32,62)(35,68)(36,65)(38,52)(41,64)(43,51)(44,55)(46,61)(47,58)(48,67)(49,69)(50,66)(54,76)(56,79)(57,75)(60,74)(63,73)(70,77)(71,80)(72,78);(1,3)(2,6)(4,11)(5,10)(7,9)(8,13)(12,16)(14,15)(17,19)(18,24)(20,29)(21,31)(22,32)(23,30)(25,44)(26,45)(27,46)(28,47)(33,53)(34,55)(35,56)(36,54)(37,59)(38,60)(39,58)(40,61)(41,57)(42,62)(43,63)(48,70)(49,71)(50,72)(51,73)(52,74)(64,75)(65,76)(66,78)(67,77)(68,79)(69,80);(3,10)(6,9)(11,16)(13,15)(17,20,23,36)(18,25,26,48)(19,29,30,54)(21,38,41,69)(22,43,39,35)(24,44,45,70)(27,51,50,68)(28,52,42,49)(31,60,57,80)(32,63,58,56)(33,64,65,37)(34,66,67,40)(46,73,72,79)(47,74,62,71)(53,75,76,59)(55,78,77,61);(2,7)(6,9)(8,14)(13,15)(19,30)(24,45)(29,54)(31,57)(32,58)(44,70)(46,72)(47,62)(53,76)(55,77)(56,63)(59,75)(60,80)(61,78)(71,74)(73,79);(1,4,5,12)(2,8,7,14)(3,11,10,16)(6,13,9,15)(18,26)(24,45)(25,48)(27,50)(28,42)(33,65)(37,64)(44,70)(46,72)(47,62)(49,52)(51,68)(53,76)(59,75)(71,74)(73,79)Polycyclic presentation:
2^9|c^2=i,e^2=f,g^2=hi,h^2=i,b^a=bf,c^a=cg,d^a=df,e^a=ei,g^a=gh,h^a=hi,c^b=cf,d^b=di,g^c=gh,h^c=hi68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7530726);