Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #465716 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C23×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #16527 |
| 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 | |
826 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)(3,9)(4,19)(5,24,26,42)(6,13)(7,12)(8,14)(10,22)(11,28,41,27)(15,43)(16,47,49,54)(17,34)(18,35)(20,31)(21,37,25,40)(23,38)(29,39)(30,33)(32,50,61,57)(36,46)(44,60,48,58)(45,59)(51,53,63,56)(52,64,55,62);(1,3,6,17,8,18,7,30)(2,9,13,34,14,35,12,33)(4,20,22,46,23,45,29,15)(5,16,27,50,26,49,28,57)(10,36,38,59,39,43,19,31)(11,32,42,47,41,61,24,54)(21,52,53,48,25,55,56,44)(37,62,63,60,40,64,51,58);(1,4,8,23)(2,10,14,39)(3,15,18,46)(5,25,26,21)(6,29,7,22)(9,31,35,59)(11,40,41,37)(12,38,13,19)(16,48,49,44)(17,45,30,20)(24,51,42,63)(27,53,28,56)(32,60,61,58)(33,36,34,43)(47,62,54,64)(50,52,57,55);(1,5,6,27,8,26,7,28)(2,11,12,24,14,41,13,42)(3,16,17,50,18,49,30,57)(4,21,22,53,23,25,29,56)(9,32,33,54,35,61,34,47)(10,37,19,51,39,40,38,63)(15,44,20,52,46,48,45,55)(31,58,43,64,59,60,36,62)Polycyclic presentation:
2^9|a^2=e,b^2=g,c^2=i,d^2=f,f^2=i,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,f^a=fi,c^b=cg,h^b=hi,d^c=de,f^c=fi,g^c=gi,e^d=ei53 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 372913);