Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #441638 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C23×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #15688 |
| 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 | |
867 conjugacy classes of subgroups.
| minimal degree | μ(G) = 64 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 64 points:
(1,2,6,13,8,12,7,14)(3,15,19,34,18,9,20,33)(4,10,24,38,23,37,30,42)(5,25,27,11,29,41,28,40)(16,31,21,35,45,43,49,58)(17,46,47,39,26,55,48,32)(22,51,53,60,52,61,54,36)(44,63,64,62,56,59,50,57);(1,3,6,19,8,18,7,20)(2,9,14,34,12,15,13,33)(4,21,24,45,23,49,30,16)(5,26,27,48,29,17,28,47)(10,35,42,31,37,58,38,43)(11,39,25,46,40,32,41,55)(22,50,53,44,52,64,54,56)(36,59,61,62,60,63,51,57);(1,4)(2,10,12,37)(3,16)(5,22)(6,30)(7,24)(8,23)(9,31,15,43)(11,36,40,60)(13,42,14,38)(17,44)(18,45)(19,49)(20,21)(25,51,41,61)(26,56)(27,54)(28,53)(29,52)(32,57,39,62)(33,58,34,35)(46,59,55,63)(47,50)(48,64);(1,5,8,29)(2,11,12,40)(3,17,18,26)(4,22,23,52)(6,28,7,27)(9,32,15,39)(10,36,37,60)(13,25,14,41)(16,44,45,56)(19,48,20,47)(21,50,49,64)(24,54,30,53)(31,57,43,62)(33,55,34,46)(35,59,58,63)(38,61,42,51)Polycyclic presentation:
2^9|a^2=g,b^2=efg,c^2=ef,d^2=i,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,f^a=fi,c^b=cg,d^b=de,e^b=ei,f^b=fi,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, 348835);