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 |
| prime radical | #60366 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
3075 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)(7,19,22,38,24,37,23,13)(8,14,27,41,28,42,34,46)(9,29,32,45,33,44,31,15)(20,39,25,53,49,62,52,35)(21,36,30,43,51,61,50,47)(26,55,57,65,58,64,56,40)(48,60,59,66,67,68,54,63);(5,7,12,24)(6,13,18,38)(8,25,28,52)(9,30,33,50)(10,23,11,22)(14,39,42,62)(15,43,45,47)(16,37,17,19)(20,34,49,27)(21,31,51,32)(26,48,58,67)(29,36,44,61)(35,46,53,41)(40,60,65,68)(54,56,59,57)(55,63,64,66);(2,4)(5,8)(6,14)(7,20)(9,26)(10,27)(11,34)(12,28)(13,35)(15,40)(16,41)(17,46)(18,42)(19,39)(21,48)(22,25)(23,52)(24,49)(29,55)(30,59)(31,56)(32,57)(33,58)(36,60)(37,62)(38,53)(43,66)(44,64)(45,65)(47,63)(50,54)(51,67)(61,68);(2,4)(5,9)(6,15)(7,21)(8,26)(10,31)(11,32)(12,33)(13,36)(14,40)(16,44)(17,29)(18,45)(19,47)(20,48)(22,50)(23,30)(24,51)(25,54)(27,56)(28,58)(34,57)(35,60)(37,43)(38,61)(39,63)(41,64)(42,65)(46,55)(49,67)(52,59)(53,68)(62,66);(5,10,12,11)(6,16,18,17)(7,22,24,23)(8,27,28,34)(9,31,33,32)(13,19,38,37)(14,41,42,46)(15,44,45,29)(20,25,49,52)(21,50,51,30)(26,56,58,57)(35,39,53,62)(36,47,61,43)(40,64,65,55)(48,54,67,59)(60,63,68,66)Polycyclic presentation:
2^9|a^2=f,b^2=i,e^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, 7408208);