Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #394202 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C27 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #2847 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| prime radical | C22×#41 | C23 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1558 conjugacy classes of subgroups.
| minimal degree | μ(G) = 68 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 4 generators on 68 points:
(5,6,11,16)(7,17,22,40)(8,23,26,13)(9,14,28,37)(10,29,18,15)(12,30,33,21)(19,43,46,38)(20,39,48,61)(24,34,53,50)(25,35,41,51)(27,54,42,36)(31,45,49,55)(32,56,58,47)(44,60,65,63)(52,67,62,59)(57,68,66,64);(1,2)(3,4)(5,7,18,30,11,22,10,21)(6,12,29,40,16,33,15,17)(8,19,41,55,26,46,25,45)(9,20,42,56,28,48,27,47)(13,31,51,43,23,49,35,38)(14,32,54,61,37,58,36,39)(24,44,62,68,53,65,52,64)(34,57,67,63,50,66,59,60);(2,4)(5,8,10,25,11,26,18,41)(6,13,15,35,16,23,29,51)(7,19,21,45,22,46,30,55)(9,24,27,52,28,53,42,62)(12,31,17,38,33,49,40,43)(14,34,36,59,37,50,54,67)(20,44,47,64,48,65,56,68)(32,57,39,60,58,66,61,63);(5,9)(6,14)(7,20)(8,24)(10,27)(11,28)(12,32)(13,34)(15,36)(16,37)(17,39)(18,42)(19,44)(21,47)(22,48)(23,50)(25,52)(26,53)(29,54)(30,56)(31,57)(33,58)(35,59)(38,60)(40,61)(41,62)(43,63)(45,64)(46,65)(49,66)(51,67)(55,68)Polycyclic presentation:
2^9|a^2=i,b^2=ei,c^2=e,e^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,c^b=cg,f^c=fi,h^c=hi,f^d=fi68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 301948);