Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #428870 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C24×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #15391 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| prime radical | C22×#41 | C23 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
915 conjugacy classes of subgroups.
| minimal degree | μ(G) = 64 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 64 points:
(1,2,6,13,8,14,7,12)(3,15,18,9,19,34,20,35)(4,21,24,38,26,37,25,10)(5,27,29,41,30,40,31,11)(16,44,46,58,22,32,47,42)(17,48,28,39,49,33,50,43)(23,36,54,60,55,61,56,51)(45,59,64,57,52,62,53,63);(1,3,6,18,8,19,7,20)(2,9,12,15,14,35,13,34)(4,22,24,47,26,16,25,46)(5,28,29,49,30,50,31,17)(10,32,37,58,38,44,21,42)(11,39,40,48,41,43,27,33)(23,53,54,45,55,64,56,52)(36,59,51,63,61,62,60,57);(1,4,8,26)(2,10,14,38)(3,16,19,22)(5,23,30,55)(6,25,7,24)(9,32,35,44)(11,36,41,61)(12,21,13,37)(15,42,34,58)(17,45,49,52)(18,47,20,46)(27,51,40,60)(28,53,50,64)(29,56,31,54)(33,57,48,63)(39,62,43,59);(1,5,8,30)(2,11)(3,17,19,49)(4,23,26,55)(6,31,7,29)(9,33)(10,36)(12,27)(13,40)(14,41)(15,43)(16,45,22,52)(18,50,20,28)(21,51)(24,56,25,54)(32,57)(34,39)(35,48)(37,60)(38,61)(42,59)(44,63)(46,53,47,64)(58,62)Polycyclic presentation:
2^9|a^2=f,b^2=e,c^2=i,d^2=ef,e^2=i,f^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,f^c=fi,h^c=hi,e^d=ei,f^d=fi,g^d=gi59 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 336067);