Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #476613 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C2×C42 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #2826 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| prime radical | C24 | 2−1+4 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
765 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)(3,4)(5,6,11,16,12,18,10,17)(7,19,24,39,23,47,22,13)(8,25,29,14,30,41,28,42)(9,31,35,15,36,45,34,46)(20,50,53,57,26,37,52,48)(21,38,56,61,32,43,55,49)(27,40,59,44,60,67,33,63)(51,68,62,64,58,65,54,66);(5,7,10,22,12,23,11,24)(6,13,16,19,18,39,17,47)(8,26,28,53,30,20,29,52)(9,32,34,56,36,21,35,55)(14,37,41,48,42,50,25,57)(15,43,45,49,46,38,31,61)(27,51,33,54,60,58,59,62)(40,66,44,68,67,64,63,65);(5,8,10,28,12,30,11,29)(6,14,16,41,18,42,17,25)(7,20,22,52,23,26,24,53)(9,33,34,60,36,59,35,27)(13,37,19,48,39,50,47,57)(15,44,45,67,46,63,31,40)(21,54,55,58,32,62,56,51)(38,65,61,66,43,68,49,64);(2,4)(5,9,12,36)(6,15,18,46)(7,21,23,32)(8,27,30,60)(10,35,11,34)(13,38,39,43)(14,40,42,67)(16,31,17,45)(19,49,47,61)(20,51,26,58)(22,56,24,55)(25,44,41,63)(28,59,29,33)(37,64,50,66)(48,68,57,65)(52,62,53,54)Polycyclic presentation:
2^9|a^2=eg,b^2=e,c^2=e,d^2=i,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=def,f^b=fi,h^b=hi,d^c=de,e^d=ei,f^d=fi,h^d=hi50 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 385525);