Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #467936 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C24 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #16595 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| prime radical | C24 | C2×#41 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
697 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)(4,21)(5,27,31,41)(6,13)(7,12)(8,14)(9,18)(10,25)(11,29,42,30)(16,44)(17,33,51,61)(19,35)(20,34)(22,36)(23,37,52,54)(24,39)(26,38)(28,55,53,40)(32,47)(43,50,60,49)(45,63,56,59)(46,58)(48,62,64,57);(1,3,6,18,8,20,7,19)(2,9,12,15,14,35,13,34)(4,22,24,16,26,46,25,47)(5,17,30,50,31,51,29,49)(10,36,38,32,39,58,21,44)(11,33,27,43,42,61,41,60)(23,48,53,45,52,64,28,56)(37,59,55,57,54,63,40,62);(1,4,6,25,8,26,7,24)(2,10,12,21,14,39,13,38)(3,16,18,22,20,47,19,46)(5,28,30,52,31,53,29,23)(9,32,15,36,35,44,34,58)(11,40,27,54,42,55,41,37)(17,48,50,56,51,64,49,45)(33,59,43,62,61,63,60,57);(1,5,6,30,8,31,7,29)(2,11,12,27,14,42,13,41)(3,17,18,50,20,51,19,49)(4,23,25,28,26,52,24,53)(9,33,15,43,35,61,34,60)(10,37,21,40,39,54,38,55)(16,45,22,48,47,56,46,64)(32,57,36,59,44,62,58,63)Polycyclic presentation:
2^9|a^2=eg,b^2=gi,c^2=f,d^2=f,e^2=i,f^2=i,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,f^a=fi,g^a=gi,c^b=cg,h^b=hi,d^c=de,e^c=ei,g^c=gi,h^c=hi,e^d=ei53 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 375133);