Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6535815 |
| 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 ◁ #2295 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1241 conjugacy classes of subgroups.
| minimal degree | μ(G) = 56 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 56 points:
(1,2,5,6)(3,7,4,8)(9,10,13,16)(11,14,17,20)(12,15,19,22)(18,21,23,24)(25,26,30,34)(27,35,38,50)(28,40,41,54)(29,43,36,33)(31,44,46,39)(32,47,48,42)(37,51,52,45)(49,55,56,53);(2,6)(7,8)(9,11)(10,14)(12,18)(13,17)(15,21)(16,20)(19,23)(22,24)(25,27,36,44,30,38,29,39)(26,31,43,50,34,46,33,35)(28,37,47,55,41,52,42,53)(32,45,54,56,48,51,40,49);(1,3,5,4)(2,7,6,8)(11,17)(14,20)(18,23)(21,24)(25,28,29,42,30,41,36,47)(26,32,33,40,34,48,43,54)(27,37,39,53,38,52,44,55)(31,45,35,49,46,51,50,56);(1,3,5,4)(2,7,6,8)(9,12)(10,15)(11,18)(13,19)(14,21)(16,22)(17,23)(20,24)(28,41)(32,48)(37,52)(40,54)(42,47)(45,51)(49,56)(53,55);(1,4,5,3)(2,8,6,7)(12,19)(15,22)(18,23)(21,24)(27,38)(31,46)(35,50)(37,52)(39,44)(45,51)(49,56)(53,55)Polycyclic presentation:
2^9|a^2=fgh,b^2=gi,c^2=f,d^2=fgi,e^2=fgi,f^2=i,g^2=i,b^a=bf,c^a=cg,f^a=fi,g^a=gi,c^b=ch,e^b=ei,d^c=di,e^d=eh74 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6476402);