Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6996719 |
| 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 | |
1718 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)(21,22,25,30,26,31,29,44)(23,32,35,42,36,34,40,27)(24,28,39,43,38,37,45,51)(33,46,48,50,47,49,52,41);(7,13)(8,16)(10,17)(11,15)(21,23,26,36)(22,27,31,42)(24,33,38,47)(25,40,29,35)(28,41,37,50)(30,34,44,32)(39,52,45,48)(43,49,51,46);(1,2,3,4)(5,7,9,13)(6,10,12,17)(8,14,16,19)(11,18,15,20)(21,24)(22,28)(23,33)(25,39)(26,38)(27,41)(29,45)(30,43)(31,37)(32,46)(34,49)(35,48)(36,47)(40,52)(42,50)(44,51);(7,13)(10,17)(14,19)(18,20)(21,22,29,44,26,31,25,30)(23,34,40,42,36,32,35,27)(24,37,45,43,38,28,39,51)(33,46,52,41,47,49,48,50);(5,8,9,16)(6,11,12,15)(7,14,13,19)(10,18,17,20)(24,38)(28,37)(33,47)(39,45)(41,50)(43,51)(46,49)(48,52)Polycyclic presentation:
2^9|a^2=fh,b^2=i,c^2=gh,d^2=fi,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=dhi,e^c=ei62 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7103982);