Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #419810 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C27 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #2668 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| prime radical | C2×#575 | C22 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1629 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)(7,19)(8,25)(9,31,35,45)(10,17)(11,16)(12,18)(13,22)(14,29)(15,34,44,33)(20,36)(21,50,52,37)(23,39)(24,38)(26,54)(27,56,59,40)(28,42)(30,41)(32,43,51,65)(46,49)(47,67,55,66)(48,64)(53,58,61,60)(57,63,62,68);(1,2)(3,4)(5,7,12,24)(6,13,18,39)(8,26,30,20)(9,32,35,51)(10,23,11,22)(14,36,42,54)(15,43,44,65)(16,38,17,19)(21,34,52,33)(25,46,41,64)(27,57,59,62)(28,48,29,49)(31,37,45,50)(40,66,56,67)(47,58,55,60)(53,68,61,63);(2,4)(5,8,11,29,12,30,10,28)(6,14,17,25,18,42,16,41)(7,20,23,49,24,26,22,48)(9,27,34,60,35,59,33,58)(13,36,38,64,39,54,19,46)(15,40,45,53,44,56,31,61)(21,47,51,57,52,55,32,62)(37,63,65,67,50,68,43,66);(5,9,10,33,12,35,11,34)(6,15,16,31,18,44,17,45)(7,21,22,32,24,52,23,51)(8,27,28,58,30,59,29,60)(13,37,19,43,39,50,38,65)(14,40,41,61,42,56,25,53)(20,47,48,62,26,55,49,57)(36,63,46,66,54,68,64,67)Polycyclic presentation:
2^9|a^2=ef,b^2=i,c^2=ei,d^2=e,e^2=i,f^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,f^a=fi,g^a=gi,c^b=cg,d^b=de,e^b=ei,f^b=fi,h^b=hi,g^c=gi,h^c=hi,f^d=fi62 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 327656);