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 | |
1335 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)(25,26,30,34)(27,35,36,48)(28,32,39,46)(29,41,43,33)(31,42,44,38)(37,49,50,56)(40,53,52,47)(45,54,55,51);(1,2)(3,7)(4,6)(5,8)(9,11,13,19)(10,14,16,17)(12,20,21,18)(15,23,24,22)(27,36)(31,44)(35,48)(37,50)(38,42)(45,55)(49,56)(51,54);(1,3)(2,5)(4,8)(6,7)(25,27,30,36)(26,31,34,44)(28,37,39,50)(29,42,43,38)(32,45,46,55)(33,48,41,35)(40,54,52,51)(47,56,53,49);(3,8)(5,7)(9,12,13,21)(10,15,16,24)(11,18,19,20)(14,22,17,23)(25,28)(26,32)(27,37)(29,40)(30,39)(31,45)(33,47)(34,46)(35,49)(36,50)(38,51)(41,53)(42,54)(43,52)(44,55)(48,56);(2,6)(5,7)(10,16)(14,17)(15,24)(22,23)(28,39)(32,46)(37,50)(40,52)(45,55)(47,53)(49,56)(51,54)Polycyclic presentation:
2^9|a^2=fi,b^2=f,c^2=i,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, 7144289);