Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #459927 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C23×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #16352 |
| 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 | |
884 conjugacy classes of subgroups.
| minimal degree | μ(G) = 64 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 64 points:
(1,2,6,14,8,12,7,13)(3,9,19,32,17,34,18,33)(4,20,23,10,25,41,24,37)(5,26,27,39,29,40,28,11)(15,42,50,57,44,35,21,30)(16,45,49,31,47,58,48,59)(22,38,53,51,54,36,55,61)(43,63,64,60,46,56,52,62);(1,3,7,19,8,17,6,18)(2,9,14,33,12,34,13,32)(4,21,24,15,25,50,23,44)(5,16,28,49,29,47,27,48)(10,35,41,57,37,42,20,30)(11,31,26,45,39,59,40,58)(22,52,55,43,54,64,53,46)(36,60,61,63,38,62,51,56);(1,4,6,23,8,25,7,24)(2,10,13,20,12,37,14,41)(3,15,18,21,17,44,19,50)(5,22,27,53,29,54,28,55)(9,30,32,42,34,57,33,35)(11,38,40,61,39,36,26,51)(16,46,48,64,47,43,49,52)(31,56,58,62,59,63,45,60);(1,5,6,27,8,29,7,28)(2,11,13,40,12,39,14,26)(3,16,18,48,17,47,19,49)(4,22,23,53,25,54,24,55)(9,31,32,58,34,59,33,45)(10,36,20,51,37,38,41,61)(15,43,21,52,44,46,50,64)(30,56,42,62,57,63,35,60)Polycyclic presentation:
2^9|a^2=ef,b^2=fi,c^2=f,d^2=f,f^2=i,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,f^a=fi,h^a=hi,c^b=cg,e^b=ei,g^b=gi,d^c=de,g^c=gi,h^c=hi,g^d=gi53 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 367124);