Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #437528 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C23×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #15602 |
| 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 | |
901 conjugacy classes of subgroups.
| minimal degree | μ(G) = 64 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 64 points:
(1,2,6,13,8,14,7,12)(3,15,19,33,18,9,20,34)(4,21,24,10,25,37,30,38)(5,26,28,39,27,40,29,11)(16,43,46,41,45,35,22,31)(17,47,48,42,50,57,49,32)(23,36,54,51,53,59,55,60)(44,56,52,58,63,61,64,62);(1,3,6,19,8,18,7,20)(2,9,12,33,14,15,13,34)(4,22,24,16,25,46,30,45)(5,17,28,48,27,50,29,49)(10,35,21,41,38,43,37,31)(11,32,40,57,39,42,26,47)(23,52,54,63,53,64,55,44)(36,58,60,56,59,62,51,61);(1,4,6,24,8,25,7,30)(2,10,12,21,14,38,13,37)(3,16,20,22,18,45,19,46)(5,23,29,55,27,53,28,54)(9,31,34,43,15,41,33,35)(11,36,26,51,39,59,40,60)(17,44,48,52,50,63,49,64)(32,56,57,62,42,61,47,58);(1,5,7,28,8,27,6,29)(2,11,13,40,14,39,12,26)(3,17,19,49,18,50,20,48)(4,23,30,54,25,53,24,55)(9,32,33,47,15,42,34,57)(10,36,37,60,38,59,21,51)(16,44,46,64,45,63,22,52)(31,56,35,58,41,61,43,62)Polycyclic presentation:
2^9|a^2=eg,b^2=f,c^2=ef,d^2=efi,f^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,g^b=gi,h^b=hi,h^c=hi,e^d=ei,f^d=fi59 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 344725);