Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #462158 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C22×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #16420 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| prime radical | C24 | 2−1+4 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
648 conjugacy classes of subgroups.
| minimal degree | μ(G) = 64 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 64 points:
(1,2,6,12,8,14,7,13)(3,15,20,9,19,34,18,35)(4,21,25,10,26,38,24,37)(5,27,29,11,30,40,31,41)(16,32,45,51,22,52,46,42)(17,33,50,57,49,62,48,43)(23,54,55,39,56,36,28,58)(44,61,47,64,53,63,59,60);(1,3,8,19)(2,9)(4,22)(5,17,30,49)(6,18,7,20)(10,32,37,52)(11,33)(12,15)(13,34)(14,35)(16,26)(21,51,38,42)(23,53)(24,46)(25,45)(27,57)(28,59)(29,48,31,50)(36,60,54,64)(39,61,58,63)(40,43)(41,62)(44,56)(47,55);(1,4,6,25,8,26,7,24)(2,10,13,21,14,37,12,38)(3,16,18,46,19,22,20,45)(5,28,29,23,30,55,31,56)(9,32,34,51,35,52,15,42)(11,39,27,54,41,58,40,36)(17,47,48,44,49,59,50,53)(33,61,57,64,62,63,43,60);(1,5,6,29,8,30,7,31)(2,11,12,40,14,41,13,27)(3,17,18,48,19,49,20,50)(4,23,24,28,26,56,25,55)(9,33,15,43,35,62,34,57)(10,36,21,39,37,54,38,58)(16,44,45,47,22,53,46,59)(32,60,51,61,52,64,42,63)Polycyclic presentation:
2^9|a^2=fg,b^2=ef,c^2=f,d^2=e,e^2=i,f^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,f^a=fi,g^a=gi,h^a=hi,c^b=cg,g^b=gi,d^c=de,e^c=ei,h^d=hi53 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 369354);