Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #396836 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C24×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #2420 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| prime radical | C24 | C2×#41 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
880 conjugacy classes of subgroups.
| minimal degree | μ(G) = 68 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 4 generators on 68 points:
(2,3)(5,6,11,17)(7,19,23,37)(8,14,29,42)(9,30,32,43)(10,18,12,16)(13,24,36,22)(15,31,44,33)(20,45,25,53)(21,35,51,61)(26,55,58,63)(27,41,28,40)(34,49,38,48)(39,56,64,57)(46,50,59,52)(47,60,54,62)(65,66,68,67);(1,2,4,3)(5,7,10,22,11,23,12,24)(6,13,16,19,17,36,18,37)(8,25,28,48,29,20,27,49)(9,21,31,50,32,51,33,52)(14,38,41,45,42,34,40,53)(15,35,43,46,44,61,30,59)(26,54,57,66,58,47,56,67)(39,62,55,65,64,60,63,68);(1,3,4,2)(5,8,10,27,11,29,12,28)(6,14,16,40,17,42,18,41)(7,20,22,48,23,25,24,49)(9,26,31,56,32,58,33,57)(13,34,19,45,36,38,37,53)(15,39,43,63,44,64,30,55)(21,47,50,66,51,54,52,67)(35,60,46,65,61,62,59,68);(5,9,10,31,11,32,12,33)(6,15,16,43,17,44,18,30)(7,21,22,50,23,51,24,52)(8,26,27,56,29,58,28,57)(13,35,19,46,36,61,37,59)(14,39,40,63,42,64,41,55)(20,47,48,66,25,54,49,67)(34,60,45,65,38,62,53,68)Polycyclic presentation:
2^9|a^2=i,b^2=fh,c^2=e,d^2=hi,e^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,h^a=hi,c^b=cg,g^b=gi,f^c=fi68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 323636);