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 ◁ #2402 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1207 conjugacy classes of subgroups.
| minimal degree | μ(G) = 56 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 56 points:
(1,2,5,8)(3,6,4,7)(9,10,13,16)(11,14,18,22)(12,15,20,24)(17,21,19,23)(25,26,30,35,31,36,29,32)(27,33,39,48,37,50,45,46)(28,40,41,47,44,34,43,52)(38,51,54,55,42,49,53,56);(1,3,5,4)(2,6,8,7)(9,11)(10,14)(12,19)(13,18)(15,23)(16,22)(17,20)(21,24)(26,32)(27,37)(28,41)(29,30)(33,48)(35,36)(38,53)(42,54)(43,44)(46,50)(47,52)(49,51);(1,3,5,4)(2,7,8,6)(9,12,13,20)(10,15,16,24)(11,17,18,19)(14,21,22,23)(25,27)(26,33)(28,42)(29,45)(30,39)(31,37)(32,46)(34,51)(35,48)(36,50)(38,44)(40,49)(41,53)(43,54)(47,56)(52,55);(1,3,5,4)(2,6,8,7)(12,20)(15,24)(17,19)(21,23)(25,28,29,43,31,44,30,41)(26,34,32,47,36,40,35,52)(27,38,45,53,37,42,39,54)(33,49,46,55,50,51,48,56);(1,4,5,3)(2,7,8,6)(10,16)(11,18)(15,24)(17,19)(25,29,31,30)(26,35,36,32)(27,39,37,45)(28,43,44,41)(33,46,50,48)(34,52,40,47)(38,54,42,53)(49,55,51,56)Polycyclic presentation:
2^9|a^2=fghi,b^2=g,c^2=ghi,d^2=fgi,e^2=gi,f^2=i,b^a=bf,c^a=cg,d^a=di,e^a=eh,c^b=ch,d^b=df,e^b=eh,f^b=fi,d^c=dh,e^c=ei68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7031126);