Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6574084 |
| 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 ◁ C2×C42 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2946 conjugacy classes of subgroups.
| minimal degree | μ(G) = 68 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 68 points:
(2,4)(5,6,12,17)(7,18,23,38)(8,14,27,44)(9,15,30,48)(10,31,11,16)(13,32,39,22)(19,40,21,50)(20,49,57,66)(24,37,52,35)(25,41,60,68)(26,43,33,42)(28,45,59,67)(29,47,34,46)(36,56,64,55)(51,62,58,63)(53,65,54,61);(5,7,10,22,12,23,11,32)(6,13,16,18,17,39,31,38)(8,24,26,19,27,52,33,21)(9,20,29,55,30,57,34,56)(14,40,42,37,44,50,43,35)(15,36,46,49,48,64,47,66)(25,53,28,51,60,54,59,58)(41,62,67,65,68,63,45,61);(1,2,3,4)(5,8)(6,14)(7,19)(9,28)(10,26)(11,33)(12,27)(13,35)(15,45)(16,42)(17,44)(18,40)(20,53)(21,23)(22,52)(24,32)(25,34)(29,60)(30,59)(31,43)(36,62)(37,39)(38,50)(41,46)(47,68)(48,67)(49,65)(51,55)(54,57)(56,58)(61,66)(63,64);(5,9)(6,15)(7,20)(8,25)(10,29)(11,34)(12,30)(13,36)(14,41)(16,46)(17,48)(18,49)(19,51)(21,58)(22,55)(23,57)(24,53)(26,28)(27,60)(31,47)(32,56)(33,59)(35,61)(37,65)(38,66)(39,64)(40,62)(42,67)(43,45)(44,68)(50,63)(52,54);(5,8)(6,14)(7,21)(9,28)(10,26)(11,33)(12,27)(13,37)(15,45)(16,42)(17,44)(18,50)(19,23)(20,54)(22,24)(25,34)(29,60)(30,59)(31,43)(32,52)(35,39)(36,63)(38,40)(41,46)(47,68)(48,67)(49,61)(51,56)(53,57)(55,58)(62,64)(65,66)Polycyclic presentation:
2^9|a^2=i,b^2=f,c^2=g,f^2=i,h^2=i,b^a=bf,c^a=cg,f^a=fi,c^b=ch,e^b=ehi,h^b=hi,d^c=dhi,h^c=hi,e^d=eh,h^d=hi,h^e=hi74 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6490400);