Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #418884 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C26×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #15267 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
2195 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)(4,10)(5,25,28,40)(6,30)(7,12)(8,13)(9,17)(11,29,38,27)(15,41)(16,32,47,56)(18,43)(19,33)(20,49)(21,51,54,57)(22,39)(23,35)(24,36)(26,42,48,37)(31,46)(34,55,61,53)(44,63,50,58)(45,59)(52,62,64,60);(1,3,6,17,8,18,7,19)(2,9,12,14,13,33,30,43)(4,20,22,46,24,15,23,45)(5,26,27,47,28,48,29,16)(10,31,35,49,36,59,39,41)(11,37,25,56,38,42,40,32)(21,52,53,44,54,64,55,50)(34,60,51,63,61,62,57,58);(1,4,8,24)(2,10,13,36)(3,15,18,20)(5,21,28,54)(6,23,7,22)(9,31,33,59)(11,34,38,61)(12,39,30,35)(14,41,43,49)(16,44,47,50)(17,46,19,45)(25,57,40,51)(26,52,48,64)(27,55,29,53)(32,58,56,63)(37,62,42,60);(1,5)(2,11)(3,16)(4,21)(6,29)(7,27)(8,28)(9,32)(10,34)(12,40)(13,38)(14,42)(15,44)(17,48)(18,47)(19,26)(20,50)(22,55)(23,53)(24,54)(25,30)(31,58)(33,56)(35,57)(36,61)(37,43)(39,51)(41,60)(45,52)(46,64)(49,62)(59,63)Polycyclic presentation:
2^9|a^2=f,b^2=e,c^2=i,e^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,g^a=gi,c^b=cg,d^b=de,g^b=gi,e^c=ei,h^c=hi,e^d=ei,f^d=fi,g^d=gi,h^d=hi59 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 326081);