Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6437602 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C211 |
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 | |
5871 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)(7,18)(8,24)(9,15)(10,33)(11,17)(12,16)(13,21)(14,27)(19,35)(20,48)(22,37)(23,45)(25,38)(26,54)(28,40)(29,46)(30,36)(31,44)(32,42)(34,43)(39,56)(41,52)(47,50)(49,59)(51,60)(53,61)(55,62)(57,64)(58,63)(65,66)(67,68);(2,4)(5,7,12,23)(6,13,16,37)(8,25,29,50)(9,30,32,53)(10,22,11,21)(14,38,40,47)(15,41,42,48)(17,45,33,18)(19,27,51,28)(20,34,52,31)(24,35,46,60)(26,49,57,67)(36,43,61,44)(39,59,63,68)(54,65,64,62)(55,58,66,56);(5,8,12,29)(6,14,16,40)(7,19,23,51)(9,26,32,57)(10,28,11,27)(13,35,37,60)(15,39,42,63)(17,46,33,24)(18,47,45,38)(20,49,52,67)(21,25,22,50)(30,55,53,66)(31,58,34,56)(36,59,61,68)(41,62,48,65)(43,64,44,54);(5,9)(6,15)(7,20)(8,26)(10,31)(11,34)(12,32)(13,36)(14,39)(16,42)(17,43)(18,48)(19,49)(21,30)(22,53)(23,52)(24,54)(25,55)(27,56)(28,58)(29,57)(33,44)(35,59)(37,61)(38,62)(40,63)(41,45)(46,64)(47,65)(50,66)(51,67)(60,68);(6,16)(13,37)(14,40)(15,42)(17,33)(18,45)(24,46)(35,60)(36,61)(38,47)(39,63)(41,48)(43,44)(54,64)(59,68)(62,65)Polycyclic presentation:
2^9|b^2=i,c^2=i,f^2=i,g^2=i,h^2=i,b^a=bf,c^a=cg,e^a=ei,f^a=fi,g^a=gi,c^b=ch,d^b=dh,f^b=fi,g^b=gi,h^b=hi,f^c=fi,g^c=gi,h^c=hi,h^d=hi80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6344801);