Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #405853 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C25 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #14919 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| prime radical | C24 | C22×D4 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
931 conjugacy classes of subgroups.
| minimal degree | μ(G) = 64 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 64 points:
(1,2,6,11,7,12,27,37)(3,13,17,40,16,8,28,31)(4,18,20,33,21,35,25,9)(5,22,23,36,24,10,26,34)(14,41,44,29,43,49,47,38)(15,30,46,56,45,39,48,57)(19,50,51,32,52,58,53,54)(42,60,62,59,61,64,63,55);(1,3)(2,8,12,13)(4,14,21,43)(5,15)(6,17)(7,16)(9,29)(10,30,22,39)(11,31,37,40)(18,49)(19,42,52,61)(20,44,25,47)(23,46)(24,45)(26,48)(27,28)(32,55)(33,38)(34,56,36,57)(35,41)(50,64)(51,62,53,63)(54,59)(58,60);(1,4)(2,9)(3,14,16,43)(5,19)(6,25)(7,21)(8,29,13,38)(10,32)(11,35)(12,33)(15,42,45,61)(17,47,28,44)(18,37)(20,27)(22,54)(23,53)(24,52)(26,51)(30,55,39,59)(31,41,40,49)(34,50)(36,58)(46,63,48,62)(56,64,57,60);(1,5,7,24)(2,10,12,22)(3,15,16,45)(4,19,21,52)(6,26,27,23)(8,30,13,39)(9,32,33,54)(11,36,37,34)(14,42,43,61)(17,48,28,46)(18,50,35,58)(20,53,25,51)(29,55,38,59)(31,57,40,56)(41,60,49,64)(44,63,47,62)Polycyclic presentation:
2^9|a^2=f,b^2=h,c^2=e,d^2=i,f^2=i,b^a=be,c^a=cf,d^a=dh,h^a=hi,c^b=cg,e^b=ei,g^b=gi,f^c=fi,g^c=gi,h^c=hi,f^d=fi65 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 313050);