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 | |
1297 conjugacy classes of subgroups.
| minimal degree | μ(G) = 56 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 56 points:
(9,10,14,18)(11,19,21,15)(12,16,13,17)(20,24,22,23)(25,26)(27,36)(28,33)(29,34)(30,32)(31,38)(35,43)(37,48)(39,46)(40,45)(41,50)(42,47)(44,52)(49,54)(51,56)(53,55);(1,2)(3,7)(4,6)(5,8)(9,11,14,21)(10,15,18,19)(12,20,13,22)(16,23,17,24)(25,27)(26,31)(28,40)(29,38)(30,39)(32,44)(33,48)(34,36)(35,46)(37,41)(42,53)(43,52)(45,50)(47,56)(49,55)(51,54);(1,3)(2,5)(4,8)(6,7)(9,12,14,13)(10,16,18,17)(11,20,21,22)(15,23,19,24)(26,32)(28,41)(30,43)(31,44)(33,49)(34,35)(36,46)(37,40)(39,52)(45,56)(47,50)(48,55);(3,8)(5,7)(9,13,14,12)(10,17,18,16)(11,20,21,22)(15,23,19,24)(25,28,29,41)(26,33,34,50)(27,37,38,40)(30,42,43,54)(31,45,36,48)(32,47,35,49)(39,51,52,53)(44,55,46,56);(2,6)(5,7)(10,18)(15,19)(16,17)(23,24)(26,34)(27,38)(28,41)(32,35)(39,52)(42,54)(45,48)(55,56)Polycyclic presentation:
2^9|a^2=fi,b^2=fi,c^2=fi,d^2=f,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, 7145442);