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, prime radical | C23 | C23×D4 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2402 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1130 conjugacy classes of subgroups.
| minimal degree | μ(G) = 56 |
| transitive degree | μt(G) = 512 |
A minimal faithful permutation representation — 5 generators on 56 points:
(1,2)(3,8)(4,6)(5,7)(9,10,13,17)(11,15,18,14)(12,20,22,16)(19,23,21,24)(25,26,30,33)(27,35,38,50)(28,40,42,32)(29,44,45,34)(31,37,47,39)(36,52,41,51)(43,49,56,55)(46,53,48,54);(9,11,13,18)(10,14,17,15)(12,21,22,19)(16,24,20,23)(25,27)(26,31)(28,41)(29,37)(30,38)(32,48)(33,47)(34,50)(35,44)(36,42)(39,45)(40,46)(43,53)(49,51)(52,55)(54,56);(1,3,5,4)(2,6,7,8)(9,11)(10,15)(12,19)(13,18)(14,17)(16,24)(20,23)(21,22)(25,28,30,42)(26,32,33,40)(27,36,38,41)(29,43,45,56)(31,46,47,48)(34,49,44,55)(35,51,50,52)(37,54,39,53);(1,4,5,3)(2,6,7,8)(11,18)(14,15)(19,21)(23,24)(26,33)(27,37)(28,42)(29,45)(31,35)(36,53)(38,39)(41,54)(46,52)(47,50)(48,51)(49,55);(9,12,13,22)(10,16,17,20)(11,19,18,21)(14,23,15,24)(27,38)(28,42)(31,47)(32,40)(35,50)(37,39)(43,56)(49,55)Polycyclic presentation:
2^9|a^2=h,b^2=hi,c^2=g,d^2=gi,e^2=hi,f^2=i,b^a=bf,c^a=cg,d^a=di,e^a=ehi,f^a=fi,c^b=ch,d^b=df,e^b=eh,f^b=fi,d^c=dh,e^c=ei,f^d=fi71 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7102020);