Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6814811 |
| 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 | |
1998 conjugacy classes of subgroups.
| minimal degree | μ(G) = 72 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 72 points:
(1,2,5,7)(3,8,4,6)(9,10)(11,22)(12,28)(13,19)(14,18)(15,21)(16,25)(17,32)(20,36)(23,51)(24,50)(26,39)(27,48)(29,37)(30,43)(31,58)(33,63)(34,44)(35,47)(38,56)(40,52)(41,62)(42,61)(45,57)(46,55)(49,54)(53,69)(59,65)(60,68)(64,70)(66,71)(67,72);(1,2)(3,6)(4,8)(5,7)(9,11)(10,16)(12,29)(13,33)(14,25)(15,26)(17,40)(18,22)(19,45)(20,39)(21,48)(23,32)(24,55)(27,36)(28,49)(30,54)(31,53)(34,57)(35,56)(37,43)(38,46)(41,65)(42,51)(44,63)(47,50)(52,61)(58,66)(59,72)(60,71)(62,64)(67,70)(68,69);(9,12,14,30)(10,17)(11,23)(13,31,34,60)(15,32,36,61)(16,37,22,49)(18,42)(19,41)(20,28)(21,43)(24,53)(25,52)(26,54)(27,29)(33,64)(35,62,55,72)(38,65,50,70)(39,40,48,51)(44,67)(45,69,63,66)(46,58)(47,68)(56,71)(57,59);(1,3,5,4)(2,6,7,8)(10,18)(12,30)(16,22)(19,44)(20,21)(23,52)(29,54)(31,60)(32,61)(38,50)(39,48)(45,63)(46,47)(53,71)(59,64)(62,72);(1,4,5,3)(2,6,7,8)(9,13,15,35,14,34,36,55)(10,19,21,47,18,44,20,46)(11,24,27,57,25,56,26,33)(12,31,32,62,30,60,61,72)(16,38,39,63,22,50,48,45)(17,41,43,68,42,67,28,58)(23,53,29,59,52,71,54,64)(37,65,40,66,49,70,51,69)Polycyclic presentation:
2^9|a^2=fi,c^2=gh,d^2=fi,e^2=fhi,g^2=i,h^2=i,b^a=bf,c^a=cg,d^a=df,g^a=gi,c^b=ch,e^b=efh,h^b=hi,d^c=di62 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6724152);