Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6993899 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C27×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C2×D42 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C2×C42 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2099 conjugacy classes of subgroups.
| minimal degree | μ(G) = 72 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 72 points:
(3,7)(6,8)(9,10)(11,23)(12,29)(13,19)(14,22)(15,21)(16,20)(17,26)(18,31)(24,51)(25,50)(27,40)(28,41)(30,58)(32,48)(33,63)(34,47)(35,45)(36,46)(37,43)(38,54)(39,56)(42,61)(44,55)(49,53)(52,70)(57,65)(59,66)(60,67)(62,68)(64,72)(69,71);(2,5)(3,7)(9,11,14,26,16,28,15,27)(10,17,21,23,20,40,22,41)(12,24,31,53,32,54,37,57)(13,33,34,55,35,56,36,25)(18,38,29,49,43,51,48,65)(19,44,46,63,45,50,47,39)(30,59,60,72,62,71,61,52)(42,66,68,70,67,69,58,64);(1,2)(3,6)(4,5)(7,8)(9,12,16,32)(10,18,20,43)(11,24,28,54)(13,30,35,62)(14,31,15,37)(17,38,40,51)(19,42,45,67)(21,29,22,48)(23,49,41,65)(25,52,55,72)(26,53,27,57)(33,59,56,71)(34,60,36,61)(39,64,63,70)(44,66,50,69)(46,68,47,58);(2,5)(6,8)(9,13)(10,19)(11,25)(12,30)(14,36)(15,34)(16,35)(17,39)(18,42)(20,45)(21,47)(22,46)(23,50)(24,52)(26,56)(27,33)(28,55)(29,58)(31,61)(32,62)(37,60)(38,64)(40,63)(41,44)(43,67)(48,68)(49,69)(51,70)(53,71)(54,72)(57,59)(65,66);(1,3,4,7)(2,6,5,8)(10,20)(17,40)(18,43)(19,45)(21,22)(23,41)(29,48)(38,51)(39,63)(42,67)(44,50)(46,47)(49,65)(58,68)(64,70)(66,69)Polycyclic presentation:
2^9|b^2=f,c^2=i,e^2=h,f^2=i,g^2=i,b^a=bf,c^a=cg,e^a=ehi,f^a=fi,g^a=gi,c^b=ch,d^b=df,e^b=eh,d^c=dh,g^c=gi,f^d=fi62 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7050831);