Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #7514172 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C29 |
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 | |
2493 conjugacy classes of subgroups.
| minimal degree | μ(G) = 68 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 68 points:
(2,3)(5,6)(7,12)(8,21)(9,26)(10,29)(11,16)(13,24)(14,27)(15,33)(17,38)(18,48)(19,36)(20,43)(22,34)(23,39)(25,44)(28,45)(30,37)(31,40)(32,42)(35,50)(41,52)(46,59)(47,61)(49,62)(51,60)(53,56)(54,68)(55,63)(57,65)(58,64)(66,67);(1,2,4,3)(5,7)(6,12)(8,22)(9,18)(10,30)(11,19)(13,38)(14,41)(15,43)(16,36)(17,24)(20,33)(21,34)(23,55)(25,51)(26,48)(27,52)(28,49)(29,37)(31,47)(32,50)(35,42)(39,63)(40,61)(44,60)(45,62)(46,56)(53,59)(54,57)(58,67)(64,66)(65,68);(5,8)(6,13)(7,17)(9,23)(10,31)(11,25)(12,34)(14,39)(15,44)(16,40)(18,46)(19,47)(20,51)(21,29)(22,30)(24,33)(26,53)(27,58)(28,57)(32,56)(35,59)(36,60)(37,61)(38,43)(41,66)(42,65)(45,64)(48,63)(49,67)(50,54)(52,55)(62,68);(5,9,11,28)(6,14,16,42)(7,18,19,49)(8,23,25,57)(10,32,33,27)(12,35,36,41)(13,39,40,65)(15,45,29,26)(17,46,47,67)(20,52,30,50)(21,53,44,64)(22,54,51,55)(24,58,31,56)(34,59,60,66)(37,62,43,48)(38,63,61,68);(1,3,4,2)(7,19)(12,36)(17,47)(18,49)(20,30)(22,51)(34,60)(35,41)(37,43)(38,61)(46,67)(48,62)(50,52)(54,55)(59,66)(63,68)Polycyclic presentation:
2^9|b^2=f,d^2=i,e^2=f,g^2=i,b^a=bf,c^a=cg,d^a=dg,e^a=ef,g^a=gi,h^a=hi,c^b=cg,d^b=dh,e^b=ei,g^b=gi,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, 7426000);