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 | C2×#121 | C23 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2396 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1340 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)(25,26,32,46,31,36,30,37)(27,38,42,33,28,34,40,49)(29,35,47,56,44,53,45,52)(39,51,54,48,41,50,43,55);(1,3,4,8)(2,5,6,7)(11,17)(12,20)(14,21)(15,19)(25,27)(26,33)(28,31)(29,43)(30,42)(32,40)(34,37)(35,51)(36,49)(38,46)(39,45)(41,47)(44,54)(48,52)(50,53)(55,56);(2,6)(5,7)(9,11,13,17)(10,14,16,21)(12,18,20,23)(15,22,19,24)(25,28,31,27)(26,34,36,38)(29,39,44,41)(30,42,32,40)(33,46,49,37)(35,51,53,50)(43,47,54,45)(48,52,55,56);(3,8)(5,7)(11,17)(14,21)(18,23)(22,24)(25,29,30,45,31,44,32,47)(26,35,37,52,36,53,46,56)(27,39,40,43,28,41,42,54)(33,48,38,51,49,55,34,50);(9,12,13,20)(10,15,16,19)(11,18,17,23)(14,22,21,24)(26,36)(27,28)(29,44)(37,46)(40,42)(45,47)(48,55)(50,51)Polycyclic presentation:
2^9|a^2=fh,b^2=g,c^2=h,d^2=fg,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, 7105116);