Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6806623 |
| 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 | #29669 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2396 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1729 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)(5,6,9,12)(7,13,15,10)(8,16,17,11)(14,18,20,19)(21,22)(23,34)(24,38)(25,40)(26,32)(27,33)(28,31)(29,37)(30,35)(36,48)(39,51)(41,52)(42,47)(43,46)(44,50)(45,49);(5,7)(6,10)(8,14)(9,15)(11,18)(12,13)(16,19)(17,20)(21,23,27,37)(22,29,33,34)(24,25,35,28)(26,39,42,49)(30,40,38,31)(32,45,47,51)(36,43,50,41)(44,46,48,52);(2,4)(21,24,28,23,27,35,25,37)(22,30,31,29,33,38,40,34)(26,39,43,36,42,49,41,50)(32,45,46,44,47,51,52,48);(5,8,9,17)(6,11,12,16)(7,14,15,20)(10,18,13,19)(21,25,27,28)(22,31,33,40)(23,35,37,24)(26,41,42,43)(29,30,34,38)(32,46,47,52)(36,49,50,39)(44,45,48,51);(7,15)(10,13)(14,20)(18,19)(21,26,25,41,27,42,28,43)(22,32,40,52,33,47,31,46)(23,36,24,39,37,50,35,49)(29,44,30,45,34,48,38,51)Polycyclic presentation:
2^9|a^2=fi,b^2=i,c^2=h,d^2=f,e^2=hi,h^2=i,b^a=bf,c^a=cg,d^a=df,c^b=ch,e^b=efh,h^b=hi,d^c=di74 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6715770);