Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 5 |
| isoclinism | isoclinic class of #13657 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C25 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C4 | #2177 |
| commutator | C16 | C23×C4 |
| Frattini, cosocle | C2×C16 | C24 |
| socle | C2 | C2×#2177 |
| prime radical | #33981 | C2 |
| LCS | G ▷ C16 ▷ C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4 ◁ C22×C4 ◁ C2×C4×C8 ◁ C2×#346 ◁ G | |
| LECS | G ▷ C2×C16 ▷ C8 ▷ C4 ▷ C2 ▷ C1 | |
421 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)(3,16,20,51)(4,14)(5,12)(6,25)(7,31)(8,33)(9,15)(10,19,37,59)(11,22)(13,28)(17,53,55,64)(18,49,34,52)(21,60)(23,46)(24,43)(26,45,54,50)(27,41)(29,40)(30,47)(32,48,58,42)(35,61,38,63)(36,56,39,62)(44,57);(1,3)(2,10)(4,17)(5,18)(6,26)(7,32)(8,34)(9,20)(11,35)(12,36)(13,42)(14,45)(15,37)(16,47)(19,57)(21,56)(22,58)(23,55)(24,61)(25,51)(27,62)(28,59)(29,63)(30,54)(31,38)(33,39)(40,49)(41,53)(43,52)(44,48)(46,50)(60,64);(1,4,8,22,9,23,5,7)(2,11,12,14,15,31,33,46)(3,17,34,58,20,55,18,32)(6,27,29,28,30,21,24,57)(10,35,36,45,37,38,39,50)(13,43,41,47,44,40,60,25)(16,48,49,64,51,42,52,53)(19,26,62,63,59,54,56,61);(1,5,9,8)(2,12,15,33)(3,18,20,34)(4,7,23,22)(6,24,30,29)(10,36,37,39)(11,14,31,46)(13,41,44,60)(16,49,51,52)(17,32,55,58)(19,56,59,62)(21,28,27,57)(25,43,47,40)(26,61,54,63)(35,45,38,50)(42,53,48,64)Polycyclic presentation:
2^9|a^2=f,c^2=fh,d^2=i,e^2=ghi,g^2=hi,h^2=i,b^a=be,c^a=ch,e^a=eg,g^a=gh,h^a=hi,e^b=egi,f^b=fi,g^b=gh,h^b=hi,e^c=ei68 conjugacy classes — view the full character table.
| maximal subgroup of | #72223646, #72223679, C2×#512778, … |
| maximal normal subgroup of | #72223646, #72223679, C2×#512778, … |
| maximal quotient of | C4×#13658, #72223673, C2×#512778, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 419974);