Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #396123 |
| 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:
(1,2,6,13,8,12,7,14)(3,15,18,34,19,9,30,33)(4,20,23,37,25,38,24,10)(5,26,27,39,29,40,28,11)(16,35,21,31,44,57,49,41)(17,45,46,42,48,55,47,32)(22,36,52,60,54,59,53,50)(43,56,62,58,63,61,51,64);(1,3)(2,9)(4,21)(5,17)(6,30)(7,18)(8,19)(10,35)(11,32)(12,15)(13,34)(14,33)(16,23)(20,41)(22,51)(24,44)(25,49)(26,55)(27,47)(28,46)(29,48)(31,38)(36,58)(37,57)(39,42)(40,45)(43,53)(50,61)(52,63)(54,62)(56,60)(59,64);(1,4,8,25)(2,10,12,37)(3,16,19,44)(5,22,29,54)(6,24,7,23)(9,31,15,41)(11,36,39,59)(13,38,14,20)(17,43,48,63)(18,49,30,21)(26,50,40,60)(27,53,28,52)(32,56,42,61)(33,35,34,57)(45,64,55,58)(46,51,47,62);(1,5)(2,11,12,39)(3,17)(4,22)(6,28)(7,27)(8,29)(9,32,15,42)(10,36,37,59)(13,40,14,26)(16,43)(18,47)(19,48)(20,50,38,60)(21,51)(23,53)(24,52)(25,54)(30,46)(31,56,41,61)(33,55,34,45)(35,58,57,64)(44,63)(49,62)Polycyclic presentation:
2^9|a^2=f,c^2=i,d^2=e,f^2=i,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,f^c=fi,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, 303320);