Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #460129 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C25×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #16357 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| prime radical | #3449 | C22 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1147 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)(5,11,25,41)(6,26)(7,12)(8,13)(9,17)(15,34)(16,47,50,43)(18,32)(19,52)(20,35,53,62)(21,37)(22,38)(23,55,54,39)(24,42,27,40)(28,33)(29,36)(30,45)(31,51,59,49)(44,58,56,61)(46,60)(48,57,63,64);(1,3,6,17,8,18,7,28)(2,9,12,14,13,33,26,32)(4,19,21,15,22,45,29,46)(5,16,27,51,25,50,24,49)(10,34,36,52,38,60,37,30)(11,31,42,43,41,59,40,47)(20,48,54,44,53,63,23,56)(35,61,55,64,62,58,39,57);(1,4)(2,10)(3,15)(5,23,25,54)(6,21)(7,29)(8,22)(9,30)(11,39,41,55)(12,36)(13,38)(14,34)(16,48,50,63)(17,45)(18,46)(19,28)(20,24,53,27)(26,37)(31,58,59,61)(32,60)(33,52)(35,40,62,42)(43,57,47,64)(44,49,56,51);(1,5,6,27,8,25,7,24)(2,11,12,42,13,41,26,40)(3,16,17,51,18,50,28,49)(4,20,21,54,22,53,29,23)(9,31,14,43,33,59,32,47)(10,35,36,55,38,62,37,39)(15,44,45,63,46,56,19,48)(30,57,34,61,52,64,60,58)Polycyclic presentation:
2^9|a^2=f,b^2=ef,c^2=f,d^2=ef,e^2=i,g^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,h^a=hi,c^b=cg,g^b=gi,h^b=hi,d^c=de,g^c=gi,e^d=ei,f^d=fi,h^d=hi53 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 367325);