Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #7234437 |
| 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 | C24 | C22×D4 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2396 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1306 conjugacy classes of subgroups.
| minimal degree | μ(G) = 56 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 56 points:
(9,10,14,17)(11,18,20,15)(12,16,13,22)(19,24,21,23)(25,26,31,36)(27,32,39,49)(28,34,41,50)(29,35,30,44)(33,43,51,40)(37,46,54,56)(38,47,42,48)(45,52,55,53);(1,2)(3,7)(4,6)(5,8)(9,11,14,20)(10,15,17,18)(12,21,13,19)(16,24,22,23)(25,27)(26,32)(28,37)(29,42)(30,38)(31,39)(33,45)(34,46)(35,48)(36,49)(40,53)(41,54)(43,52)(44,47)(50,56)(51,55);(1,3)(2,5)(4,8)(6,7)(25,28,31,41)(26,33,36,51)(27,37,39,54)(29,40,30,43)(32,45,49,55)(34,35,50,44)(38,52,42,53)(46,48,56,47);(3,8)(5,7)(9,12,14,13)(10,16,17,22)(11,19,20,21)(15,23,18,24)(25,28)(26,34)(27,37)(29,43)(30,40)(31,41)(32,46)(33,44)(35,51)(36,50)(38,53)(39,54)(42,52)(45,47)(48,55)(49,56);(2,6)(5,7)(9,13,14,12)(10,16,17,22)(11,19,20,21)(15,24,18,23)(25,29,31,30)(26,35,36,44)(27,38,39,42)(28,40,41,43)(32,47,49,48)(33,50,51,34)(37,52,54,53)(45,46,55,56)Polycyclic presentation:
2^9|a^2=fi,b^2=f,c^2=i,d^2=f,e^2=fi,g^2=i,b^a=bf,c^a=cg,e^a=ef,g^a=gi,c^b=ch,d^b=df,e^b=ehi,d^c=dhi,g^c=gi,e^d=ei56 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7149638);