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 | |
1353 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)(25,26,30,35)(27,31,38,47)(28,33,39,49)(29,41,43,34)(32,42,48,40)(36,45,50,56)(37,52,54,46)(44,53,55,51);(1,2)(3,7)(4,6)(5,8)(9,11,13,19)(10,14,16,17)(12,20,21,18)(15,23,24,22)(25,27,30,38)(26,31,35,47)(28,36,39,50)(29,37,43,54)(32,44,48,55)(33,45,49,56)(34,46,41,52)(40,51,42,53);(1,3)(2,5)(4,8)(6,7)(25,28)(26,32)(27,36)(29,42)(30,39)(31,44)(33,41)(34,49)(35,48)(37,53)(38,50)(40,43)(45,52)(46,56)(47,55)(51,54);(3,8)(5,7)(9,12,13,21)(10,15,16,24)(11,18,19,20)(14,22,17,23)(25,28)(26,33)(27,36)(29,40)(30,39)(31,45)(32,41)(34,48)(35,49)(37,51)(38,50)(42,43)(44,52)(46,55)(47,56)(53,54);(2,6)(5,7)(10,16)(14,17)(15,24)(22,23)(28,39)(32,48)(33,49)(36,50)(40,42)(44,55)(45,56)(51,53)Polycyclic presentation:
2^9|a^2=fi,b^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=eh,d^c=dh,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, 7144248);