Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6996639 |
| 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 | |
1374 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)(27,36)(28,39)(29,34)(30,35)(31,32)(33,40)(37,49)(38,50)(41,55)(42,46)(43,47)(44,56)(45,52)(48,54)(51,53);(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)(28,40,42,37)(29,43,45,38)(33,39,49,46)(34,50,52,47)(41,53,44,54)(48,56,51,55);(1,2,5,4)(3,6,8,7)(9,12)(10,15)(11,17)(13,20)(14,21)(16,24)(18,19)(22,23)(25,28,31,40,30,42,27,37)(26,33,36,39,35,49,32,46)(29,44,43,53,45,41,38,54)(34,51,50,56,52,48,47,55);(1,4,5,2)(3,6,8,7)(12,20)(15,24)(17,19)(21,23)(25,29,30,45)(26,34,35,52)(27,38,31,43)(28,41,42,44)(32,47,36,50)(33,48,49,51)(37,53,40,54)(39,55,46,56);(10,16)(11,18)(15,24)(17,19)(26,35)(28,42)(29,45)(32,36)(37,40)(38,43)(48,51)(55,56)Polycyclic presentation:
2^9|a^2=fh,b^2=i,c^2=fgi,d^2=f,g^2=i,b^a=bf,c^a=cg,e^a=eh,g^a=gi,c^b=ch,d^b=df,e^b=ehi,d^c=dh,e^c=ei,e^d=ei68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6951587);