Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6994139 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C28 |
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 | |
1667 conjugacy classes of subgroups.
| minimal degree | μ(G) = 52 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 52 points:
(1,2,3,4)(8,16)(11,17)(15,20)(18,19)(21,22,28,34)(23,35,25,31)(24,38,41,30)(26,33,27,32)(29,42,45,37)(36,49,39,47)(40,48,43,50)(44,52,46,51);(5,6,9,12)(7,13,14,10)(8,17,16,11)(15,18,20,19)(21,23,27,37,28,25,26,42)(22,29,33,35,34,45,32,31)(24,39,40,51,41,36,43,52)(30,46,48,49,38,44,50,47);(2,4)(5,7)(6,10)(8,15)(9,14)(11,18)(12,13)(16,20)(17,19)(21,24,28,41)(22,30,34,38)(23,36,25,39)(26,43,27,40)(29,44,45,46)(31,49,35,47)(32,50,33,48)(37,52,42,51);(7,14)(10,13)(15,20)(18,19)(21,25,28,23)(22,31,34,35)(24,36,41,39)(26,42,27,37)(29,33,45,32)(30,47,38,49)(40,51,43,52)(44,50,46,48);(5,8,9,16)(6,11,12,17)(7,15,14,20)(10,18,13,19)(21,26,28,27)(22,32,34,33)(23,37,25,42)(24,40,41,43)(29,35,45,31)(30,48,38,50)(36,51,39,52)(44,49,46,47)Polycyclic presentation:
2^9|a^2=g,b^2=fhi,c^2=i,d^2=i,e^2=h,f^2=i,b^a=bf,c^a=cg,e^a=eh,f^a=fi,c^b=ch,d^b=dfi,e^b=eh,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, 7089923);