Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #451628 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C24 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #16005 |
| 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 | |
737 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)(3,15,19,9)(4,20)(5,25)(6,14)(7,13)(8,12)(10,23)(11,27)(16,31,44,42)(17,46,48,58)(18,33,30,41)(21,50,45,34)(22,53)(24,36)(26,38)(28,39)(29,40)(32,47,37,49)(35,57)(43,61,63,64)(51,56)(52,60,54,62)(55,59);(1,3)(2,9)(4,21,23,45)(5,17)(6,18)(7,30)(8,19)(10,34,20,50)(11,37)(12,15)(13,33)(14,41)(16,24,44,29)(22,54,55,52)(25,58)(26,47)(27,49)(28,48)(31,36,42,40)(32,38)(35,61,51,64)(39,46)(43,56,63,57)(53,62,59,60);(1,4)(2,10,12,20)(3,16)(5,22)(6,29)(7,24)(8,23)(9,31,15,42)(11,35,38,51)(13,40,14,36)(17,43)(18,45)(19,44)(21,30)(25,59,39,53)(26,57)(27,56)(28,55)(32,60,37,62)(33,50,41,34)(46,64,58,61)(47,52)(48,63)(49,54);(1,5,8,28)(2,11,12,38)(3,17,19,48)(4,22,23,55)(6,27,7,26)(9,32,15,37)(10,35,20,51)(13,25,14,39)(16,43,44,63)(18,49,30,47)(21,52,45,54)(24,57,29,56)(31,60,42,62)(33,46,41,58)(34,61,50,64)(36,53,40,59)Polycyclic presentation:
2^9|a^2=fgh,b^2=f,c^2=ef,d^2=i,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,g^a=gi,h^a=hi,c^b=cg,d^b=de,h^b=hi,e^c=ei,f^c=fi,g^c=gi,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, 358825);