Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #419810 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C24×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #2668 |
| 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 | |
1135 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)(5,6,12,16,11,18,10,17)(7,19,22,13,24,45,23,36)(8,25,32,14,28,39,27,40)(9,15,30,43,31,44,33,42)(20,34,48,46,50,37,49,62)(21,51,52,35,53,41,29,58)(26,54,56,66,57,65,60,38)(47,63,59,64,67,61,55,68);(5,7,10,23,11,24,12,22)(6,13,17,19,18,36,16,45)(8,20,27,49,28,50,32,48)(9,29,30,21,31,52,33,53)(14,37,25,46,40,34,39,62)(15,41,43,58,44,51,42,35)(26,55,56,47,57,59,60,67)(38,64,54,61,66,68,65,63);(5,8,11,28)(6,14,18,40)(7,20,24,50)(9,26,31,57)(10,32,12,27)(13,34,36,37)(15,38,44,66)(16,25,17,39)(19,46,45,62)(21,47,53,67)(22,49,23,48)(29,59,52,55)(30,60,33,56)(35,61,58,63)(41,64,51,68)(42,54,43,65);(2,4)(5,9,11,31)(6,15)(7,21)(8,26,28,57)(10,30,12,33)(13,35,36,58)(14,38)(16,42)(17,43)(18,44)(19,41,45,51)(20,47)(22,29)(23,52)(24,53)(25,54)(27,56,32,60)(34,61,37,63)(39,65)(40,66)(46,64,62,68)(48,55)(49,59)(50,67)Polycyclic presentation:
2^9|a^2=eg,b^2=f,c^2=i,d^2=ef,e^2=i,f^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,g^a=gi,h^a=hi,c^b=cg,d^b=de,e^b=ei,g^b=gi,h^b=hi,e^c=ei,f^c=fi,g^d=gi,h^d=hi62 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 343903);