Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #25798 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C26×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #2433 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C23×C4 | C24 |
| prime radical | C22×#135 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1794 conjugacy classes of subgroups.
| minimal degree | μ(G) = 68 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 4 generators on 68 points:
(1,2,4,3)(5,6)(7,17)(8,23)(9,28)(10,31)(11,16)(12,21)(13,27)(14,29)(15,18)(19,38)(20,46)(22,40)(24,52)(25,55)(26,36)(30,59)(32,45)(33,48)(34,41)(35,61)(37,43)(39,47)(42,51)(44,65)(49,54)(50,56)(53,60)(57,58)(62,66)(63,64)(67,68);(1,3,4,2)(5,7,18,41,11,22,10,21)(6,12,31,40,16,34,15,17)(8,24,42,45,27,19,26,54)(9,20,43,64,30,48,29,47)(13,32,51,52,23,49,36,38)(14,33,59,63,37,46,28,39)(25,53,61,66,57,44,56,68)(35,60,55,67,50,65,58,62);(5,8)(6,13,16,23)(7,19,22,24)(9,25)(10,26)(11,27)(12,32)(14,35,37,50)(15,36,31,51)(17,38)(18,42)(20,44,48,53)(21,45,41,54)(28,58,59,55)(29,56)(30,57)(33,60)(34,49)(39,62)(40,52)(43,61)(46,65)(47,66,64,68)(63,67);(5,9,18,43,11,30,10,29)(6,14,31,59,16,37,15,28)(7,20,41,64,22,48,21,47)(8,25,42,61,27,57,26,56)(12,33,40,63,34,46,17,39)(13,35,51,55,23,50,36,58)(19,44,54,68,24,53,45,66)(32,60,52,67,49,65,38,62)Polycyclic presentation:
2^9|a^2=h,b^2=ehi,c^2=fg,d^2=ei,e^2=i,b^a=be,c^a=cf,d^a=de,e^a=ei,g^a=gi,c^b=cg,f^b=fi,f^c=fi,g^c=gi80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 276393);