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 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2001 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)(3,5)(4,6)(7,8)(9,10,16,22)(11,23,27,46)(12,28,30,48)(13,19,34,45)(14,21,15,20)(17,35,40,26)(18,37,42,31)(24,38,52,65)(25,50,56,43)(29,58,60,68)(32,63,55,39)(33,47,36,44)(41,61,67,62)(49,53,57,54)(51,64,70,72)(59,69,71,66);(1,2)(3,7)(4,6)(5,8)(9,11)(10,17)(12,24)(13,32)(14,35)(15,26)(16,27)(18,38)(19,43)(20,46)(21,23)(22,40)(25,33)(28,57)(29,59)(30,52)(31,54)(34,55)(36,56)(37,53)(39,44)(41,66)(42,65)(45,50)(47,63)(48,49)(51,61)(58,72)(60,71)(62,70)(64,68)(67,69);(1,3,4,8)(2,5,6,7)(9,12,16,30)(10,18,22,42)(11,24,27,52)(13,29,34,60)(14,31,15,37)(17,38,40,65)(19,41,45,67)(20,28,21,48)(23,49,46,57)(25,51,56,70)(26,53,35,54)(32,59,55,71)(33,61,36,62)(39,64,63,72)(43,66,50,69)(44,68,47,58);(3,8)(5,7)(9,13,16,34)(10,19,22,45)(11,25,27,56)(12,29,30,60)(14,36,15,33)(17,39,40,63)(18,41,42,67)(20,47,21,44)(23,50,46,43)(24,51,52,70)(26,55,35,32)(28,58,48,68)(31,62,37,61)(38,64,65,72)(49,69,57,66)(53,71,54,59);(2,6)(5,7)(12,30)(18,42)(24,52)(28,48)(29,60)(31,37)(38,65)(41,67)(49,57)(51,70)(53,54)(58,68)(59,71)(61,62)(64,72)(66,69)Polycyclic presentation:
2^9|a^2=i,c^2=hi,d^2=i,f^2=i,g^2=i,b^a=bf,c^a=cg,e^a=eh,f^a=fi,g^a=gi,c^b=ch,d^b=df,e^b=eh,f^b=fi,d^c=dh,e^c=ei,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, 6962133);