Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6993899 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C28 |
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 | |
1778 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)(9,10,15,21,16,20,14,35)(11,22,26,40,27,50,25,17)(12,28,30,42,31,43,37,18)(13,19,33,47,34,45,36,46)(23,38,52,57,53,65,56,48)(24,49,32,63,54,44,55,39)(29,58,61,67,62,68,60,41)(51,64,70,66,71,72,59,69);(2,5)(3,7)(9,11,14,25,16,27,15,26)(10,17,21,22,20,40,35,50)(12,23,30,52,31,53,37,56)(13,32,33,54,34,55,36,24)(18,38,43,48,42,65,28,57)(19,44,46,63,45,49,47,39)(29,59,60,71,62,70,61,51)(41,66,58,72,67,69,68,64);(1,2)(3,6)(4,5)(7,8)(9,12,16,31)(10,18,20,42)(11,23,27,53)(13,29,34,62)(14,30,15,37)(17,38,40,65)(19,41,45,67)(21,43,35,28)(22,48,50,57)(24,51,54,71)(25,52,26,56)(32,59,55,70)(33,60,36,61)(39,64,63,72)(44,66,49,69)(46,58,47,68);(2,5)(6,8)(9,13)(10,19)(11,24)(12,29)(14,36)(15,33)(16,34)(17,39)(18,41)(20,45)(21,47)(22,49)(23,51)(25,55)(26,32)(27,54)(28,58)(30,61)(31,62)(35,46)(37,60)(38,64)(40,63)(42,67)(43,68)(44,50)(48,69)(52,70)(53,71)(56,59)(57,66)(65,72);(1,3,4,7)(2,6,5,8)(10,20)(17,40)(18,42)(19,45)(21,35)(22,50)(28,43)(38,65)(39,63)(41,67)(44,49)(46,47)(48,57)(58,68)(64,72)(66,69)Polycyclic presentation:
2^9|a^2=g,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,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, 7056607);