Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6996355 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C27 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C23×D4 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| prime radical | C22×D4 | C24 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2399 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1325 conjugacy classes of subgroups.
| minimal degree | μ(G) = 56 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 56 points:
(2,6)(7,8)(12,20)(15,21)(19,24)(22,23)(25,26,31,35)(27,32,30,45)(28,38,41,55)(29,34,44,51)(33,36,48,40)(37,46,43,50)(39,54,42,56)(47,52,49,53);(1,2)(3,7)(4,8)(5,6)(9,10,13,16)(11,17,18,14)(12,21,20,15)(19,22,24,23)(25,27,31,30)(26,32,35,45)(28,36,41,40)(29,37,44,43)(33,38,48,55)(34,46,51,50)(39,52,42,53)(47,54,49,56);(1,3,5,4)(2,7,6,8)(9,11)(10,14)(12,19)(13,18)(15,22)(16,17)(20,24)(21,23)(25,28,30,40,31,41,27,36)(26,33,32,38,35,48,45,55)(29,42,43,52,44,39,37,53)(34,49,46,56,51,47,50,54);(1,4,5,3)(2,7,6,8)(11,18)(14,17)(19,24)(22,23)(25,29,31,44)(26,34,35,51)(27,37,30,43)(28,39,41,42)(32,46,45,50)(33,47,48,49)(36,52,40,53)(38,54,55,56);(9,12,13,20)(10,15,16,21)(11,19,18,24)(14,22,17,23)(25,27,31,30)(26,32,35,45)(28,36,41,40)(29,43,44,37)(33,38,48,55)(34,50,51,46)(39,53,42,52)(47,56,49,54)Polycyclic presentation:
2^9|a^2=i,b^2=hi,c^2=fg,d^2=fi,e^2=hi,g^2=i,b^a=bf,c^a=cg,e^a=eh,g^a=gi,c^b=ch,d^b=df,e^b=eh,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, 7094156);