Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #472953 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C22×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #16842 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| prime radical | C24 | 2−1+4 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
646 conjugacy classes of subgroups.
| minimal degree | μ(G) = 64 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 64 points:
(1,2,6,11,7,12,29,38)(3,13,16,32,17,40,27,8)(4,18,21,41,22,9,28,35)(5,23,25,57,26,10,36,39)(14,42,45,30,19,52,50,33)(15,46,48,31,49,43,59,55)(20,34,24,37,54,51,58,56)(44,61,47,62,53,63,64,60);(1,3,6,27,7,17,29,16)(2,8,11,40,12,32,38,13)(4,19,21,45,22,14,28,50)(5,15,25,59,26,49,36,48)(9,33,35,52,18,30,41,42)(10,31,39,46,23,55,57,43)(20,53,24,47,54,44,58,64)(34,62,37,61,51,60,56,63);(1,4,7,22)(2,9,12,18)(3,14,17,19)(5,24,26,58)(6,28,29,21)(8,30,32,33)(10,37,23,56)(11,41,38,35)(13,42,40,52)(15,47,49,64)(16,50,27,45)(20,36,54,25)(31,61,55,63)(34,57,51,39)(43,60,46,62)(44,59,53,48);(1,5,6,25,7,26,29,36)(2,10,38,57,12,23,11,39)(3,15,27,59,17,49,16,48)(4,20,21,24,22,54,28,58)(8,31,13,43,32,55,40,46)(9,34,41,56,18,51,35,37)(14,44,50,64,19,53,45,47)(30,60,52,61,33,62,42,63)Polycyclic presentation:
2^9|a^2=e,b^2=egh,c^2=i,d^2=ef,e^2=i,b^a=be,c^a=cf,d^a=dh,f^a=fi,c^b=cg,e^b=ei,f^b=fi,g^b=gi,d^c=de,e^c=ei,g^d=gi,h^d=hi53 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 380150);