Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6996375 |
| 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 | #392 | D4 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2392 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1251 conjugacy classes of subgroups.
| minimal degree | μ(G) = 56 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 56 points:
(1,2,5,4)(3,7,8,6)(9,10,13,16)(11,14,18,22)(12,15,20,24)(17,21,19,23)(25,26,30,35)(27,36,31,32)(28,39,41,45)(29,34,44,50)(33,37,48,42)(38,49,43,46)(40,52,54,55)(47,51,56,53);(1,3)(2,6)(4,7)(5,8)(9,11)(10,14)(12,19)(13,18)(15,23)(16,22)(17,20)(21,24)(25,27,30,31)(26,32,35,36)(28,37,41,42)(29,43,44,38)(33,45,48,39)(34,49,50,46)(40,53,54,51)(47,52,56,55);(1,2,5,4)(3,6,8,7)(9,12)(10,15)(11,17)(13,20)(14,21)(16,24)(18,19)(22,23)(25,28,27,37,30,41,31,42)(26,33,32,45,35,48,36,39)(29,40,38,51,44,54,43,53)(34,47,46,55,50,56,49,52);(1,2,5,4)(3,7,8,6)(12,20)(15,24)(17,19)(21,23)(25,29,30,44)(26,34,35,50)(27,38,31,43)(28,40,41,54)(32,46,36,49)(33,47,48,56)(37,51,42,53)(39,52,45,55);(1,4,5,2)(3,7,8,6)(10,16)(11,18)(15,24)(17,19)(29,44)(34,50)(38,43)(40,54)(46,49)(47,56)(51,53)(52,55)Polycyclic presentation:
2^9|a^2=fh,b^2=i,c^2=fg,d^2=f,e^2=fi,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=dh,e^d=ei74 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6959584);