Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #448887 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C23×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #15924 |
| 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 | |
806 conjugacy classes of subgroups.
| minimal degree | μ(G) = 64 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 64 points:
(1,2,8,12,7,14,6,13)(3,15,20,34,19,41,18,9)(4,21,24,37,26,10,25,42)(5,27,29,40,30,39,31,11)(16,44,46,35,22,43,50,32)(17,47,28,38,48,33,49,64)(23,53,57,36,56,62,55,51)(45,60,54,61,52,59,58,63);(1,3,7,19)(2,9)(4,22)(5,28)(6,18,8,20)(10,35,21,32)(11,38,40,64)(12,15)(13,41)(14,34)(16,26)(17,31)(23,54,56,58)(24,50)(25,46)(27,33,39,47)(29,48)(30,49)(36,61)(37,44,42,43)(45,57,52,55)(51,63)(53,60)(59,62);(1,4,8,25,7,26,6,24)(2,10,13,42,14,21,12,37)(3,16,20,50,19,22,18,46)(5,23,31,57,30,56,29,55)(9,32,41,44,34,35,15,43)(11,36,27,53,40,51,39,62)(17,45,49,54,48,52,28,58)(33,59,64,61,47,60,38,63);(1,5,6,29,7,30,8,31)(2,11,12,39,14,40,13,27)(3,17,20,49,19,48,18,28)(4,23,24,55,26,56,25,57)(9,33,41,64,34,47,15,38)(10,36,37,62,21,51,42,53)(16,45,50,54,22,52,46,58)(32,59,44,61,35,60,43,63)Polycyclic presentation:
2^9|a^2=ef,b^2=f,c^2=ei,d^2=eg,e^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,f^a=fi,c^b=cg,d^b=de,g^b=gi,h^b=hi,f^c=fi,h^c=hi,f^d=fi59 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 356084);