Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #479768 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C22×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #17239 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| prime radical | C24 | H(5,𝔽2) |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
599 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)(3,14,18,34)(4,19)(5,25)(6,11)(7,13)(8,17,35,52)(9,22)(10,28)(12,31)(15,32,20,36)(16,48,26,33)(21,54)(23,39)(24,38)(27,58)(29,41)(30,42)(37,56)(40,55)(43,47,60,46)(44,51,62,50)(45,61,53,64)(49,63,57,59);(1,3,7,18)(2,8)(4,20)(5,26,29,16)(6,17,31,52)(9,36,39,32)(10,33)(11,14)(12,34)(13,35)(15,24)(19,43,38,60)(21,45)(22,47)(23,46)(25,44)(27,49)(28,51,30,50)(37,63,54,59)(40,64,58,61)(41,62)(42,48)(53,56)(55,57);(1,4,6,22,7,24,31,23)(2,9,12,19,13,39,11,38)(3,15,17,46,18,20,52,47)(5,27,28,56,29,55,30,21)(8,32,34,60,35,36,14,43)(10,40,25,37,42,58,41,54)(16,49,50,53,26,57,51,45)(33,61,44,59,48,64,62,63);(1,5,7,29)(2,10,13,42)(3,16,18,26)(4,21,24,56)(6,30,31,28)(8,33,35,48)(9,37,39,54)(11,25,12,41)(14,44,34,62)(15,45,20,53)(17,51,52,50)(19,40,38,58)(22,55,23,27)(32,59,36,63)(43,64,60,61)(46,57,47,49)Polycyclic presentation:
2^9|a^2=g,b^2=eh,c^2=f,d^2=i,e^2=i,f^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,f^a=fi,h^a=hi,c^b=cg,d^b=def,g^b=gi,d^c=de,h^c=hi,e^d=ei,f^d=fi,h^d=hi47 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 386965);