Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #253459 |
| 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 | |
752 conjugacy classes of subgroups.
| minimal degree | μ(G) = 64 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 64 points:
(1,2,7,13)(3,9,17,34)(4,19,22,50)(5,11,26,42)(6,27,29,41)(8,14,25,12)(10,35,37,23)(15,44,20,52)(16,31,49,61)(18,33,48,43)(21,39,56,38)(24,40,55,51)(28,32,58,62)(30,53,60,45)(36,57,63,47)(46,59,54,64);(1,3)(2,9)(4,20,22,15)(5,16)(6,28)(7,17)(8,18)(10,36,37,63)(11,31)(12,43)(13,34)(14,33)(19,44,50,52)(21,53,56,45)(23,57,35,47)(24,54,55,46)(25,48)(26,49)(27,32)(29,58)(30,38,60,39)(40,59,51,64)(41,62)(42,61);(1,4,8,24)(2,10,14,39)(3,15,18,46)(5,23,6,21)(7,22,25,55)(9,30,33,36)(11,40,27,19)(12,38,13,37)(16,47,28,45)(17,20,48,54)(26,35,29,56)(31,44,32,59)(34,60,43,63)(41,50,42,51)(49,57,58,53)(52,62,64,61);(1,5,25,29)(2,11,12,41)(3,16,48,58)(4,21,55,35)(6,7,26,8)(9,31,43,62)(10,19,38,51)(13,42,14,27)(15,45,54,57)(17,49,18,28)(20,53,46,47)(22,56,24,23)(30,59,63,52)(32,34,61,33)(36,44,60,64)(37,50,39,40)Polycyclic presentation:
2^9|a^2=h,b^2=e,c^2=i,d^2=hi,f^2=i,b^a=be,c^a=cf,d^a=de,f^a=fi,g^a=ghi,c^b=cg,f^b=fi,g^b=gh,d^c=di,e^c=eh,f^c=fi,f^d=fh44 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 160684);