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 | C2×#121 | C23 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2396 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1342 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)(25,26,31,32,30,36,29,35)(27,33,43,45,38,48,39,49)(28,40,44,50,42,34,41,46)(37,51,54,56,53,47,52,55);(1,3,4,8)(2,5,6,7)(11,17)(12,20)(14,21)(15,19)(26,32)(28,41)(29,31)(33,45)(35,36)(37,52)(39,43)(42,44)(46,50)(48,49)(53,54)(55,56);(2,6)(5,7)(9,11,13,17)(10,14,16,21)(12,18,20,23)(15,22,19,24)(25,27)(26,33)(28,37)(29,39)(30,38)(31,43)(32,45)(34,47)(35,49)(36,48)(40,51)(41,52)(42,53)(44,54)(46,55)(50,56);(3,8)(5,7)(11,17)(14,21)(18,23)(22,24)(25,28,29,41,30,42,31,44)(26,34,35,50,36,40,32,46)(27,37,39,52,38,53,43,54)(33,47,49,56,48,51,45,55);(9,12,13,20)(10,15,16,19)(11,18,17,23)(14,22,21,24)(27,38)(33,48)(37,53)(39,43)(45,49)(47,51)(52,54)(55,56)Polycyclic presentation:
2^9|a^2=fhi,b^2=g,c^2=h,d^2=fg,e^2=h,f^2=i,b^a=bf,c^a=cg,d^a=di,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, 7105004);