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)(11,23,27,48)(12,28,30,50)(13,19,35,47)(14,21,15,20)(17,36,41,26)(18,38,43,31)(24,39,54,69)(25,52,59,44)(29,61,33,67)(32,65,58,40)(34,49,37,46)(42,63,45,64)(51,55,60,56)(53,68,57,70)(62,72,66,71);(1,2)(3,7)(4,6)(5,8)(9,11)(10,17)(12,24)(13,32)(14,36)(15,26)(16,27)(18,39)(19,44)(20,48)(21,23)(22,41)(25,34)(28,60)(29,62)(30,54)(31,56)(33,66)(35,58)(37,59)(38,55)(40,46)(42,71)(43,69)(45,72)(47,52)(49,65)(50,51)(53,63)(57,64)(61,70)(67,68);(1,3,4,8)(2,5,6,7)(9,12,16,30)(10,18,22,43)(11,24,27,54)(13,33,35,29)(14,31,15,38)(17,39,41,69)(19,45,47,42)(20,28,21,50)(23,51,48,60)(25,57,59,53)(26,55,36,56)(32,66,58,62)(34,64,37,63)(40,70,65,68)(44,72,52,71)(46,61,49,67);(3,8)(5,7)(9,13,16,35)(10,19,22,47)(11,25,27,59)(12,29,30,33)(14,37,15,34)(17,40,41,65)(18,42,43,45)(20,49,21,46)(23,52,48,44)(24,53,54,57)(26,58,36,32)(28,61,50,67)(31,64,38,63)(39,68,69,70)(51,72,60,71)(55,66,56,62);(2,6)(5,7)(12,30)(18,43)(24,54)(28,50)(29,33)(31,38)(39,69)(42,45)(51,60)(53,57)(55,56)(61,67)(62,66)(63,64)(68,70)(71,72)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=dhi,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, 6962165);