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 | |
1671 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)(23,34)(24,30)(25,31)(26,32)(27,43)(28,33)(29,38)(35,50)(36,49)(37,46)(39,51)(40,47)(41,45)(42,48)(44,52);(5,6,9,12)(7,13,14,10)(8,17,16,11)(15,18,20,19)(21,23,26,37,28,39,27,38)(22,29,43,51,33,46,32,34)(24,35,42,36,25,41,40,52)(30,44,47,45,31,49,48,50);(2,4)(5,7)(6,10)(8,15)(9,14)(11,18)(12,13)(16,20)(17,19)(21,24,28,25)(22,30,33,31)(23,35,39,41)(26,42,27,40)(29,44,46,49)(32,48,43,47)(34,50,51,45)(36,38,52,37);(7,14)(10,13)(15,20)(18,19)(21,25,28,24)(22,31,33,30)(23,36,39,52)(26,42,27,40)(29,45,46,50)(32,48,43,47)(34,49,51,44)(35,38,41,37);(5,8,9,16)(6,11,12,17)(7,15,14,20)(10,18,13,19)(21,26,28,27)(22,32,33,43)(23,37,39,38)(24,40,25,42)(29,34,46,51)(30,47,31,48)(35,52,41,36)(44,45,49,50)Polycyclic presentation:
2^9|a^2=g,b^2=fhi,c^2=i,d^2=i,e^2=hi,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, 7089924);