Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6574077 |
| 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 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2490 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)(5,6,12,17)(7,18,23,40)(8,14,27,46)(9,15,32,52)(10,33,11,16)(13,34,41,22)(19,42,21,54)(20,53,28,63)(24,39,56,37)(25,43,62,68)(26,45,35,44)(29,48,30,49)(31,51,36,50)(38,59,47,58)(55,67,60,65)(57,64,61,66);(5,7,11,34,12,23,10,22)(6,13,33,40,17,41,16,18)(8,24,35,21,27,56,26,19)(9,28,36,58,32,20,31,59)(14,42,45,37,46,54,44,39)(15,47,51,53,52,38,50,63)(25,57,29,55,62,61,30,60)(43,65,49,66,68,67,48,64);(2,4)(5,8)(6,14)(7,19)(9,29)(10,26)(11,35)(12,27)(13,37)(15,48)(16,44)(17,46)(18,42)(20,57)(21,23)(22,56)(24,34)(25,31)(28,61)(30,32)(33,45)(36,62)(38,65)(39,41)(40,54)(43,51)(47,67)(49,52)(50,68)(53,64)(55,59)(58,60)(63,66);(5,9)(6,15)(7,20)(8,25)(10,31)(11,36)(12,32)(13,38)(14,43)(16,50)(17,52)(18,53)(19,55)(21,60)(22,58)(23,28)(24,61)(26,30)(27,62)(29,35)(33,51)(34,59)(37,64)(39,66)(40,63)(41,47)(42,67)(44,48)(45,49)(46,68)(54,65)(56,57);(5,8)(6,14)(7,21)(9,30)(10,26)(11,35)(12,27)(13,39)(15,49)(16,44)(17,46)(18,54)(19,23)(20,57)(22,24)(25,36)(28,61)(29,32)(31,62)(33,45)(34,56)(37,41)(38,65)(40,42)(43,50)(47,67)(48,52)(51,68)(53,64)(55,59)(58,60)(63,66)Polycyclic presentation:
2^9|a^2=i,b^2=fi,f^2=i,h^2=i,b^a=bf,c^a=cg,f^a=fi,c^b=ch,d^b=di,e^b=ehi,h^b=hi,d^c=dh,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, 6481277);