Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #430117 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C23×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #15413 |
| 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 | |
910 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)(3,15,18,9,20,41,19,33)(4,10,30,36,24,38,23,37)(5,25,29,42,28,11,27,40)(16,34,45,31,46,43,21,49)(17,39,26,32,47,55,48,57)(22,51,54,61,53,35,52,60)(44,58,63,56,64,62,50,59);(1,3,8,18,7,20,6,19)(2,9,12,41,14,33,13,15)(4,21,30,16,24,45,23,46)(5,26,29,47,28,48,27,17)(10,34,36,31,38,43,37,49)(11,39,40,32,25,55,42,57)(22,44,54,63,53,64,52,50)(35,59,60,58,51,56,61,62);(1,4,7,24)(2,10,14,38)(3,16,20,46)(5,22,28,53)(6,30,8,23)(9,31,33,49)(11,35,25,51)(12,37,13,36)(15,43,41,34)(17,44,47,64)(18,21,19,45)(26,50,48,63)(27,54,29,52)(32,56,57,59)(39,62,55,58)(40,61,42,60);(1,5,7,28)(2,11)(3,17)(4,22,24,53)(6,27,8,29)(9,32,33,57)(10,35)(12,40)(13,42)(14,25)(15,39,41,55)(16,44)(18,26)(19,48)(20,47)(21,50)(23,52,30,54)(31,56,49,59)(34,58,43,62)(36,60)(37,61)(38,51)(45,63)(46,64)Polycyclic presentation:
2^9|a^2=ef,b^2=ef,c^2=i,d^2=f,e^2=i,g^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,f^a=fi,c^b=cg,d^b=de,e^b=ei,f^b=fi,h^b=hi,e^c=ei,g^c=gi,g^d=gi,h^d=hi59 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 337314);