Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6996355 |
| 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 | |
1322 conjugacy classes of subgroups.
| minimal degree | μ(G) = 56 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 56 points:
(1,2,4,6)(3,7,8,5)(9,10,13,16)(11,14,17,21)(12,19,20,15)(18,24,23,22)(25,26,29,34,30,35,41,31)(27,32,38,46,37,45,51,42)(28,33,39,47,40,48,52,43)(36,44,49,55,50,54,56,53);(1,3,4,8)(2,5,6,7)(11,17)(12,20)(14,21)(15,19)(26,31)(28,39)(29,41)(32,42)(34,35)(36,49)(38,51)(40,52)(43,47)(45,46)(50,56)(53,55);(2,6)(5,7)(9,11,13,17)(10,14,16,21)(12,18,20,23)(15,22,19,24)(25,27)(26,32)(28,36)(29,38)(30,37)(31,42)(33,44)(34,46)(35,45)(39,49)(40,50)(41,51)(43,53)(47,55)(48,54)(52,56);(3,8)(5,7)(11,17)(14,21)(18,23)(22,24)(25,28,29,39,30,40,41,52)(26,33,34,47,35,48,31,43)(27,36,38,49,37,50,51,56)(32,44,46,55,45,54,42,53);(9,12,13,20)(10,15,16,19)(11,18,17,23)(14,22,21,24)(27,37)(32,45)(36,50)(38,51)(42,46)(44,54)(49,56)(53,55)Polycyclic presentation:
2^9|a^2=fh,b^2=g,c^2=h,d^2=fg,e^2=h,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=ei68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7104987);