Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #254539 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C25 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #2404 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| prime radical | C25 | C2×D4 |
| LCS | G ▷ C23×C4 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C22 ▷ C1 | |
747 conjugacy classes of subgroups.
| minimal degree | μ(G) = 64 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 64 points:
(4,18)(5,23)(8,27)(9,30)(11,37)(13,38)(16,45)(21,52)(22,49)(24,46)(25,26)(28,31)(29,56)(33,35)(34,61)(36,39)(40,64)(41,63)(44,53)(48,54)(50,62)(51,55)(57,58)(59,60);(1,2,6,12)(3,13)(4,9,18,30)(5,11,25,39)(7,10,20,32)(8,24)(14,40)(15,38)(16,29)(17,41)(19,50)(21,34,48,59)(22,35,49,33)(23,37,26,36)(27,45)(28,55)(31,53)(42,63)(43,64)(44,57)(46,56)(47,62)(51,58)(52,61,54,60);(1,3,7,17)(2,8,10,27)(4,19,22,14)(5,24,26,45)(6,15,20,42)(9,31,35,28)(11,38,36,63)(12,29,32,56)(13,37,41,39)(16,23,46,25)(18,47,49,43)(21,53,52,55)(30,58,33,57)(34,62,61,64)(40,59,50,60)(44,48,51,54);(1,4,20,49)(2,9,32,33)(3,14,42,47)(5,21,23,54)(6,18,7,22)(8,28,56,58)(10,35,12,30)(11,34,37,60)(13,40,63,62)(15,43,17,19)(16,44,45,53)(24,55,46,51)(25,48,26,52)(27,31,29,57)(36,61,39,59)(38,64,41,50)Polycyclic presentation:
2^9|b^2=eh,c^2=i,d^2=hi,g^2=h,b^a=be,c^a=cf,d^a=deh,g^a=ghi,c^b=cg,f^b=fi,d^c=di,e^c=eh,g^c=gh,f^d=fh44 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 161746);