Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6996719 |
| 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 ◁ #2399 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1271 conjugacy classes of subgroups.
| minimal degree | μ(G) = 56 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 56 points:
(2,5)(6,7)(9,10,13,16)(11,14,17,21)(12,19,20,15)(18,24,23,22)(25,26,30,32)(27,37,38,48)(28,34,42,46)(29,36,31,35)(33,40,45,41)(39,51,52,56)(43,50,44,49)(47,54,55,53);(1,2,4,5)(3,7,8,6)(11,17)(12,20)(14,21)(15,19)(26,32)(27,38)(28,42)(35,36)(40,41)(43,44)(47,55)(51,56);(9,11,13,17)(10,14,16,21)(12,18,20,23)(15,22,19,24)(25,27,29,40,30,38,31,41)(26,33,35,48,32,45,36,37)(28,39,44,54,42,52,43,53)(34,47,50,56,46,55,49,51);(1,3,4,8)(2,6,5,7)(11,17)(14,21)(18,23)(22,24)(25,28,30,42)(26,34,32,46)(27,39,38,52)(29,44,31,43)(33,47,45,55)(35,50,36,49)(37,51,48,56)(40,54,41,53);(9,12,13,20)(10,15,16,19)(11,18,17,23)(14,22,21,24)(25,29,30,31)(26,35,32,36)(27,40,38,41)(28,43,42,44)(33,48,45,37)(34,49,46,50)(39,53,52,54)(47,51,55,56)Polycyclic presentation:
2^9|a^2=h,b^2=fi,c^2=gh,d^2=f,e^2=h,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=dhi,e^d=ei62 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7104207);