Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #394590 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C27 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #2622 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| prime radical | C22×2−1+4 | C22 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
2078 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)(7,17,21,40)(8,22,26,13)(9,14,28,39)(10,29,31,15)(12,30,34,20)(18,43,23,41)(19,42,47,62)(24,36,53,48)(25,37,54,49)(27,55,57,38)(32,45,35,51)(33,56,59,46)(44,63,50,64)(52,66,68,61)(58,67,60,65);(1,2)(3,4)(5,7,11,21)(6,12,16,34)(8,23,26,18)(9,19,28,47)(10,30,31,20)(13,35,22,32)(14,33,39,59)(15,40,29,17)(24,50,53,44)(25,45,54,51)(27,56,57,46)(36,60,48,58)(37,41,49,43)(38,62,55,42)(52,65,68,67)(61,63,66,64);(2,4)(5,8,10,25,11,26,31,54)(6,13,15,37,16,22,29,49)(7,18,20,45,21,23,30,51)(9,24,27,52,28,53,57,68)(12,32,17,41,34,35,40,43)(14,36,38,61,39,48,55,66)(19,44,46,65,47,50,56,67)(33,58,42,63,59,60,62,64);(5,9)(6,14)(7,19)(8,24)(10,27)(11,28)(12,33)(13,36)(15,38)(16,39)(17,42)(18,44)(20,46)(21,47)(22,48)(23,50)(25,52)(26,53)(29,55)(30,56)(31,57)(32,58)(34,59)(35,60)(37,61)(40,62)(41,63)(43,64)(45,65)(49,66)(51,67)(54,68)Polycyclic presentation:
2^9|a^2=i,b^2=i,c^2=e,e^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,c^b=cg,e^b=ei,f^c=fi,g^c=gi,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, 301789);