Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #7501009 |
| 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 |
| prime radical | #31075 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C2×C42 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1744 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)(3,4)(5,7)(6,8)(9,10,14,21,16,20,15,22)(11,17,25,41,26,40,37,48)(12,27,30,44,32,45,31,18)(13,33,34,19,35,47,36,46)(23,42,28,56,50,67,55,38)(24,51,52,39,54,66,53,65)(29,59,60,57,61,43,62,63)(49,69,70,68,71,64,58,72);(1,3)(2,6)(4,7)(5,8)(9,11,16,26)(10,17,20,40)(12,28,32,55)(13,24,35,54)(14,25,15,37)(18,42,44,67)(19,39,46,65)(21,41,22,48)(23,30,50,31)(27,56,45,38)(29,58,61,70)(33,51,47,66)(34,52,36,53)(43,68,59,72)(49,62,71,60)(57,69,63,64);(9,12,16,32)(10,18,20,44)(11,23,26,50)(13,29,35,61)(14,31,15,30)(17,38,40,56)(19,43,46,59)(21,45,22,27)(24,49,54,71)(25,55,37,28)(33,63,47,57)(34,62,36,60)(39,64,65,69)(41,67,48,42)(51,72,66,68)(52,58,53,70);(2,7)(3,8)(9,13)(10,19)(11,24)(12,29)(14,36)(15,34)(16,35)(17,39)(18,43)(20,46)(21,33)(22,47)(23,49)(25,53)(26,54)(27,57)(28,58)(30,62)(31,60)(32,61)(37,52)(38,64)(40,65)(41,51)(42,68)(44,59)(45,63)(48,66)(50,71)(55,70)(56,69)(67,72);(1,4,5,6)(2,3,7,8)(10,20)(17,40)(18,44)(19,46)(21,22)(27,45)(33,47)(38,56)(39,65)(41,48)(42,67)(43,59)(51,66)(57,63)(64,69)(68,72)Polycyclic presentation:
2^9|a^2=g,b^2=i,c^2=i,e^2=f,g^2=i,h^2=i,b^a=bf,c^a=cg,d^a=dfgi,e^a=ei,c^b=ch,d^b=df,h^b=hi,g^c=gi,h^c=hi,g^d=gi68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7411157);