Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6437798 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C28 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C22×#121 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| prime radical | C22×#41 | C23 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C2×#117 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1851 conjugacy classes of subgroups.
| minimal degree | μ(G) = 72 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 72 points:
(1,2)(3,6)(4,5)(7,8)(9,10,14,20,15,21,16,22)(11,23,26,41,27,42,28,17)(12,29,33,49,32,18,37,45)(13,19,35,47,36,48,38,50)(24,51,30,39,53,43,55,64)(25,46,34,62,54,65,56,40)(31,57,60,68,59,44,61,67)(52,69,58,63,70,66,71,72);(9,11,15,27)(10,17,21,41)(12,30,32,55)(13,34,36,56)(14,28,16,26)(18,43,29,51)(19,46,48,65)(20,42,22,23)(24,37,53,33)(25,38,54,35)(31,52,59,70)(39,49,64,45)(40,50,62,47)(44,63,57,72)(58,60,71,61)(66,67,69,68);(2,5)(6,8)(9,12)(10,18)(11,24)(13,31)(14,33)(15,32)(16,37)(17,39)(19,44)(20,45)(21,29)(22,49)(23,43)(25,52)(26,30)(27,53)(28,55)(34,58)(35,60)(36,59)(38,61)(40,63)(41,64)(42,51)(46,66)(47,67)(48,57)(50,68)(54,70)(56,71)(62,72)(65,69);(9,13)(10,19)(11,25)(12,31)(14,35)(15,36)(16,38)(17,40)(18,44)(20,47)(21,48)(22,50)(23,46)(24,52)(26,34)(27,54)(28,56)(29,57)(30,58)(32,59)(33,60)(37,61)(39,63)(41,62)(42,65)(43,66)(45,67)(49,68)(51,69)(53,70)(55,71)(64,72);(1,3,4,7)(2,6,5,8)(12,32)(18,29)(24,53)(30,55)(31,59)(33,37)(39,64)(43,51)(44,57)(45,49)(52,70)(58,71)(60,61)(63,72)(66,69)(67,68)Polycyclic presentation:
2^9|a^2=f,b^2=i,e^2=gi,f^2=i,h^2=i,b^a=bf,c^a=cg,c^b=ch,d^b=dh,f^b=fi,h^b=hi,e^c=ei,h^c=hi,h^d=hi80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6361609);