Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #7234485 |
| 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×#41 | C23 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2396 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1357 conjugacy classes of subgroups.
| minimal degree | μ(G) = 56 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 56 points:
(9,10,13,16)(11,17,19,14)(12,15,21,24)(18,23,20,22)(25,26)(27,37)(28,41)(29,34)(30,35)(31,46)(32,39)(33,42)(36,45)(38,51)(40,49)(43,48)(44,50)(47,56)(52,54)(53,55);(1,2)(3,7)(4,6)(5,8)(9,11,13,19)(10,14,16,17)(12,20,21,18)(15,23,24,22)(25,27,30,39)(26,32,35,37)(28,38,29,43)(31,40,45,55)(33,47,50,52)(34,51,41,48)(36,49,46,53)(42,54,44,56);(1,3)(2,5)(4,8)(6,7)(25,28)(26,33)(27,38)(29,30)(31,44)(32,47)(34,36)(35,50)(37,52)(39,43)(40,56)(41,46)(42,45)(48,53)(49,51)(54,55);(3,8)(5,7)(9,12,13,21)(10,15,16,24)(11,18,19,20)(14,22,17,23)(25,29,30,28)(26,34,35,41)(27,38,39,43)(31,44,45,42)(32,48,37,51)(33,46,50,36)(40,54,55,56)(47,49,52,53);(2,6)(5,7)(10,16)(14,17)(15,24)(22,23)(26,35)(27,39)(28,29)(36,46)(40,55)(42,44)(47,52)(48,51)Polycyclic presentation:
2^9|a^2=fi,b^2=f,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,e^c=ei,g^c=gi,e^d=ei62 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7144327);