Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6500456 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C26×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C23×D4 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| prime radical | C24 | C22×D4 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2296 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1324 conjugacy classes of subgroups.
| minimal degree | μ(G) = 56 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 56 points:
(1,2,4,5)(3,7,8,6)(9,10,13,16)(11,14,17,20)(12,15,19,22)(18,21,23,24)(25,26)(27,35)(28,33)(29,42)(30,32)(31,39)(34,36)(37,48)(38,47)(40,45)(41,53)(43,49)(44,52)(46,50)(51,55)(54,56);(2,5)(6,7)(9,11)(10,14)(12,18)(13,17)(15,21)(16,20)(19,23)(22,24)(25,27,36,49,30,38,29,39)(26,31,42,47,32,43,34,35)(28,37,50,56,40,51,41,52)(33,44,53,55,45,54,46,48);(1,3,4,8)(2,6,5,7)(11,17)(12,19)(14,20)(15,22)(26,32)(28,40)(31,43)(34,42)(35,47)(37,51)(41,50)(52,56);(9,12,13,19)(10,15,16,22)(11,18,17,23)(14,21,20,24)(25,28,30,40)(26,33,32,45)(27,37,38,51)(29,41,36,50)(31,44,43,54)(34,46,42,53)(35,48,47,55)(39,52,49,56);(11,17)(14,20)(18,23)(21,24)(27,38)(28,40)(31,43)(33,45)(35,47)(39,49)(41,50)(46,53)Polycyclic presentation:
2^9|a^2=ghi,b^2=fg,c^2=gi,d^2=hi,f^2=i,b^a=bf,c^a=cg,f^a=fi,c^b=ch,e^b=ehi,d^c=dhi,e^d=ei68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6435818);