Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6996639 |
| 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×D4 | C24 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2402 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1243 conjugacy classes of subgroups.
| minimal degree | μ(G) = 56 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 56 points:
(1,2,5,7)(3,6,4,8)(10,14)(11,17)(12,20)(22,24)(25,26,32,48,31,36,30,37)(27,38,47,33,40,53,29,35)(28,41,49,52,44,34,45,56)(39,55,43,51,42,54,46,50);(1,3,5,4)(2,6,7,8)(9,10,13,14)(11,18,17,15)(12,21,20,16)(19,22,23,24)(25,27)(26,33)(28,42)(29,32)(30,47)(31,40)(34,51)(35,36)(37,53)(38,48)(39,44)(41,50)(43,49)(45,46)(52,54)(55,56);(1,4,5,3)(2,6,7,8)(9,11,13,17)(10,15,14,18)(12,19,20,23)(16,22,21,24)(25,28,31,44)(26,34,36,41)(27,39,40,42)(29,46,47,43)(30,45,32,49)(33,50,35,51)(37,52,48,56)(38,54,53,55);(10,14)(11,17)(16,21)(19,23)(25,29,32,27,31,47,30,40)(26,35,48,38,36,33,37,53)(28,43,49,42,44,46,45,39)(34,50,56,55,41,51,52,54);(9,12,13,20)(10,16,14,21)(11,19,17,23)(15,22,18,24)(26,36)(27,40)(28,44)(29,47)(37,48)(45,49)(50,51)(54,55)Polycyclic presentation:
2^9|a^2=fg,b^2=gi,c^2=g,d^2=fh,e^2=hi,f^2=i,b^a=bf,c^a=cg,e^a=eh,c^b=ch,d^b=df,e^b=eh,f^b=fi,d^c=dh,e^c=ei,e^d=ei68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7107155);