Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6996639 |
| 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 | C22×D8 | C23 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2399 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1730 conjugacy classes of subgroups.
| minimal degree | μ(G) = 52 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 52 points:
(2,4)(5,6,9,12)(7,10,13,17)(8,15,16,11)(14,20,19,18)(21,22,28,42,27,32,26,33)(23,34,38,29,24,30,36,45)(25,31,43,52,40,49,41,48)(35,47,50,44,37,46,39,51);(7,13)(8,16)(10,17)(11,15)(21,23)(22,29)(24,27)(25,39)(26,38)(28,36)(30,33)(31,47)(32,45)(34,42)(35,41)(37,43)(40,50)(44,48)(46,49)(51,52);(1,2,3,4)(5,7,9,13)(6,10,12,17)(8,14,16,19)(11,18,15,20)(21,24,27,23)(22,30,32,34)(25,35,40,37)(26,38,28,36)(29,42,45,33)(31,47,49,46)(39,43,50,41)(44,48,51,52);(7,13)(10,17)(14,19)(18,20)(21,25,26,41,27,40,28,43)(22,31,33,48,32,49,42,52)(23,35,36,39,24,37,38,50)(29,44,34,47,45,51,30,46);(5,8,9,16)(6,11,12,15)(7,14,13,19)(10,18,17,20)(22,32)(23,24)(25,40)(33,42)(36,38)(41,43)(44,51)(46,47)Polycyclic presentation:
2^9|a^2=fh,c^2=gh,d^2=f,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, 7103999);