Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6500535 |
| 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 | C2×#121 | C23 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2296 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1293 conjugacy classes of subgroups.
| minimal degree | μ(G) = 56 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 56 points:
(1,2,5,7)(3,8,4,6)(9,10,13,16)(11,14,18,22)(12,15,20,24)(17,21,19,23)(25,26,30,32)(27,36,39,52)(28,34,43,47)(29,35,31,45)(33,38,46,40)(37,53,41,56)(42,50,44,51)(48,55,49,54);(1,3,5,4)(2,6,7,8)(9,11)(10,14)(12,19)(13,18)(15,23)(16,22)(17,20)(21,24)(26,32)(33,46)(34,47)(35,45)(36,52)(48,49)(50,51)(53,56);(9,12)(10,15)(11,17)(13,20)(14,21)(16,24)(18,19)(22,23)(25,27,31,40,30,39,29,38)(26,33,35,36,32,46,45,52)(28,41,42,55,43,37,44,54)(34,49,51,56,47,48,50,53);(12,20)(15,24)(17,19)(21,23)(25,28,30,43)(26,34,32,47)(27,37,39,41)(29,44,31,42)(33,48,46,49)(35,51,45,50)(36,53,52,56)(38,55,40,54);(1,4,5,3)(2,6,7,8)(11,18)(14,22)(17,19)(21,23)(25,29,30,31)(26,35,32,45)(27,38,39,40)(28,42,43,44)(33,36,46,52)(34,50,47,51)(37,54,41,55)(48,56,49,53)Polycyclic presentation:
2^9|a^2=fh,b^2=fi,c^2=g,d^2=i,e^2=f,g^2=i,b^a=bf,c^a=cg,g^a=gi,c^b=ch,e^b=eh,d^c=dhi,e^d=ei74 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6419736);