Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #417980 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C24×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C4 | #2670 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| socle | C2 | #15418 |
| prime radical | #23412 | C2 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1137 conjugacy classes of subgroups.
| minimal degree | μ(G) = 64 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 64 points:
(1,2,6,12,8,13,7,28)(3,9,16,32,17,33,18,46)(4,19,23,35,22,36,27,10)(5,24,25,11,26,37,29,38)(14,39,41,56,20,49,45,30)(15,42,43,31,44,57,47,58)(21,51,53,60,52,34,54,48)(40,59,62,64,50,55,63,61);(1,3,6,16,8,17,7,18)(2,9,12,32,13,33,28,46)(4,20,23,45,22,14,27,41)(5,15,25,43,26,44,29,47)(10,30,19,39,35,56,36,49)(11,31,37,57,38,58,24,42)(21,40,53,62,52,50,54,63)(34,59,48,64,51,55,60,61);(1,4)(2,10)(3,14)(5,21)(6,27)(7,23)(8,22)(9,30)(11,34)(12,36)(13,35)(15,40)(16,45)(17,20)(18,41)(19,28)(24,48)(25,54)(26,52)(29,53)(31,55)(32,49)(33,56)(37,60)(38,51)(39,46)(42,61)(43,63)(44,50)(47,62)(57,64)(58,59);(1,5,6,25,8,26,7,29)(2,11,12,37,13,38,28,24)(3,15,16,43,17,44,18,47)(4,21,27,54,22,52,23,53)(9,31,32,57,33,58,46,42)(10,34,36,60,35,51,19,48)(14,40,45,63,20,50,41,62)(30,55,49,64,56,59,39,61)Polycyclic presentation:
2^9|a^2=f,b^2=f,d^2=ef,f^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,g^a=gi,h^a=hi,c^b=cg,d^b=de,g^b=gi,e^c=ei,f^c=fi,g^d=gi,h^d=hi62 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 337509);