Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6994139 |
| 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 | |
1192 conjugacy classes of subgroups.
| minimal degree | μ(G) = 56 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 56 points:
(1,2,5,4)(3,7,8,6)(9,10,14,17)(11,18,12,15)(13,21,22,16)(19,23,20,24)(25,26,30,35)(27,31,38,45)(28,33,39,47)(29,32,41,34)(36,43,49,54)(37,42,51,44)(40,46,52,48)(50,53,56,55);(1,3,5,8)(2,6,4,7)(9,11)(10,15)(12,14)(13,20)(16,24)(17,18)(19,22)(21,23)(25,27,30,38)(26,31,35,45)(28,36,39,49)(29,37,41,51)(32,42,34,44)(33,43,47,54)(40,50,52,56)(46,53,48,55);(1,4,5,2)(3,7,8,6)(9,12,14,11)(10,15,17,18)(13,20,22,19)(16,23,21,24)(26,32)(29,41)(31,42)(33,46)(34,35)(37,51)(40,52)(43,53)(44,45)(47,48)(50,56)(54,55);(3,8)(6,7)(11,12)(15,18)(19,20)(23,24)(25,28)(26,33)(27,36)(29,40)(30,39)(31,43)(32,46)(34,48)(35,47)(37,50)(38,49)(41,52)(42,53)(44,55)(45,54)(51,56);(9,13,14,22)(10,16,17,21)(11,19,12,20)(15,23,18,24)(28,39)(33,47)(36,49)(40,52)(43,54)(46,48)(50,56)(53,55)Polycyclic presentation:
2^9|a^2=fi,b^2=fhi,c^2=f,e^2=h,g^2=i,b^a=bf,c^a=cg,e^a=eh,g^a=gi,c^b=ch,d^b=df,e^b=eh,d^c=dh,g^c=gi,e^d=ei71 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7087505);