Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #469778 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C24×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #16662 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| prime radical | C2×#575 | C22 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1011 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)(4,10)(5,26)(6,14)(7,12)(8,13)(9,18)(11,28)(16,44)(17,48)(19,34)(20,33)(21,42)(22,39)(23,37)(24,38)(25,36)(27,35)(29,41)(30,40)(31,47)(32,51)(43,50)(45,58)(46,59)(49,64)(52,61)(53,63)(54,62)(55,56)(57,60);(1,3,6,18,8,19,7,20)(2,9,12,15,13,33,14,34)(4,21,23,47,24,16,25,46)(5,17,28,50,29,51,30,52)(10,31,36,42,38,59,37,44)(11,32,26,43,40,48,41,61)(22,53,27,57,54,45,55,49)(35,58,39,60,56,63,62,64);(1,4,8,24)(2,10,13,38)(3,16)(5,27)(6,23,7,25)(9,31)(11,39)(12,36,14,37)(15,42)(17,49,51,57)(18,46)(19,21)(20,47)(22,30)(26,56)(28,54)(29,55)(32,60,48,64)(33,59)(34,44)(35,41)(40,62)(43,63,61,58)(45,50,53,52);(1,5,6,28,8,29,7,30)(2,11,14,41,13,40,12,26)(3,17,18,50,19,51,20,52)(4,22,25,55,24,54,23,27)(9,32,34,61,33,48,15,43)(10,35,36,39,38,56,37,62)(16,45,47,57,21,53,46,49)(31,58,42,60,59,63,44,64)Polycyclic presentation:
2^9|b^2=e,c^2=fi,d^2=eg,e^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,f^b=fi,d^c=de,g^c=gi,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, 376975);