Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6682596 |
| 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×#469 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2296 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1737 conjugacy classes of subgroups.
| minimal degree | μ(G) = 52 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 52 points:
(5,6)(7,13)(8,12)(9,11)(10,15)(14,20)(16,18)(17,19)(21,22)(23,28)(24,35)(25,30)(26,40)(27,31)(29,36)(32,47)(33,42)(34,44)(37,45)(38,43)(39,46)(41,50)(48,51)(49,52);(1,2)(3,4)(5,7,9,15)(6,10,11,13)(8,16,17,14)(12,20,19,18)(21,23,27,34)(22,28,31,44)(24,32,37,49)(25,38,26,33)(29,41,46,51)(30,43,40,42)(35,47,45,52)(36,50,39,48);(2,4)(21,24,26,36,27,37,25,39)(22,29,30,35,31,46,40,45)(23,32,33,50,34,49,38,48)(28,41,43,47,44,51,42,52);(6,11)(10,13)(12,19)(18,20)(21,25,27,26)(22,30,31,40)(23,33,34,38)(24,36,37,39)(28,42,44,43)(29,45,46,35)(32,48,49,50)(41,47,51,52);(2,4)(5,8,9,17)(6,12,11,19)(7,14,15,16)(10,18,13,20)(24,37)(29,46)(32,49)(35,45)(36,39)(41,51)(47,52)(48,50)Polycyclic presentation:
2^9|b^2=fi,c^2=g,d^2=i,e^2=f,g^2=i,b^a=bf,c^a=cg,d^a=df,g^a=gi,c^b=ch,d^b=di,e^b=efh,d^c=di,e^c=ei68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6593130);