Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #396104 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C26 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #14730 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| prime radical | C2×#575 | C22 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1344 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)(3,14,17,9)(4,10)(5,24,25,56)(6,28)(7,13)(8,12)(11,27,38,26)(15,30,43,40)(16,44)(18,32,29,33)(19,48,47,34)(20,50,51,55)(21,39)(22,37)(23,36)(31,45)(35,53,60,52)(41,57)(42,62)(46,54)(49,61)(58,64)(59,63);(1,3)(2,9)(4,19)(5,16)(6,29)(7,18)(8,17)(10,34)(11,31)(12,14)(13,33)(15,21)(20,49)(22,43)(23,47)(24,54)(25,46)(26,57)(27,45)(28,32)(30,37)(35,59)(36,48)(38,41)(39,40)(42,52)(44,56)(50,58)(51,64)(53,63)(55,61)(60,62);(1,4,8,23)(2,10,12,36)(3,15,17,43)(5,20,25,51)(6,22,7,21)(9,30,14,40)(11,35,38,60)(13,39,28,37)(16,42,46,63)(18,47,29,19)(24,55,56,50)(26,53,27,52)(31,58,41,61)(32,48,33,34)(44,62,54,59)(45,64,57,49);(1,5)(2,11,12,38)(3,16)(4,20)(6,26)(7,27)(8,25)(9,31,14,41)(10,35,36,60)(13,24,28,56)(15,42)(17,46)(18,45)(19,49)(21,52)(22,53)(23,51)(29,57)(30,58,40,61)(32,44,33,54)(34,59,48,62)(37,50,39,55)(43,63)(47,64)Polycyclic presentation:
2^9|a^2=f,c^2=i,d^2=e,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,g^a=gi,c^b=cg,f^b=fi,g^b=gi,g^c=gi,h^c=hi,f^d=fi65 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 303301);