Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #7501009 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C28×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 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
3071 conjugacy classes of subgroups.
| minimal degree | μ(G) = 68 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 68 points:
(1,2)(3,4)(5,6,10,16,12,18,11,17)(7,19,23,37,22,38,24,13)(8,14,27,41,28,42,34,46)(9,29,31,44,32,45,33,15)(20,39,25,53,49,62,52,35)(21,36,30,43,50,61,51,47)(26,55,56,65,58,64,57,40)(48,60,59,66,67,68,54,63);(5,7)(6,13)(8,25)(9,30)(10,24)(11,23)(12,22)(14,39)(15,43)(16,38)(17,19)(18,37)(20,27)(21,31)(26,48)(28,52)(29,36)(32,51)(33,50)(34,49)(35,41)(40,60)(42,62)(44,47)(45,61)(46,53)(54,56)(55,63)(57,59)(58,67)(64,66)(65,68);(2,4)(5,8,12,28)(6,14,18,42)(7,20,22,49)(9,26,32,58)(10,27,11,34)(13,35,37,53)(15,40,44,65)(16,41,17,46)(19,39,38,62)(21,48,50,67)(23,25,24,52)(29,55,45,64)(30,59,51,54)(31,56,33,57)(36,60,61,68)(43,66,47,63);(2,4)(5,9)(6,15)(7,21)(8,26)(10,33)(11,31)(12,32)(13,36)(14,40)(16,45)(17,29)(18,44)(19,47)(20,48)(22,50)(23,51)(24,30)(25,54)(27,57)(28,58)(34,56)(35,60)(37,61)(38,43)(39,63)(41,64)(42,65)(46,55)(49,67)(52,59)(53,68)(62,66);(7,22)(13,37)(19,38)(20,49)(21,50)(23,24)(25,52)(30,51)(35,53)(36,61)(39,62)(43,47)(48,67)(54,59)(60,68)(63,66)Polycyclic presentation:
2^9|a^2=f,c^2=i,f^2=i,h^2=i,b^a=bf,c^a=cg,d^a=dfg,c^b=ch,d^b=df,e^b=ei,f^b=fi,h^b=hi,h^c=hi,f^d=fi68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7408206);