Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6682440 |
| 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×D4 | C2×D4 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2293 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1647 conjugacy classes of subgroups.
| minimal degree | μ(G) = 52 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 52 points:
(5,6,9,12)(7,13,15,10)(8,16,17,11)(14,18,20,19)(21,22)(23,28)(24,34)(25,30)(26,41)(27,31)(29,36)(32,44)(33,42)(35,50)(37,51)(38,45)(39,52)(40,47)(43,49)(46,48);(1,2)(3,4)(5,7)(6,10)(8,14)(9,15)(11,18)(12,13)(16,19)(17,20)(21,23,27,26)(22,28,31,41)(24,32,37,36)(25,38,40,33)(29,34,44,51)(30,45,47,42)(35,49,39,48)(43,52,46,50);(2,4)(21,24,26,36,27,37,23,32)(22,29,28,34,31,44,41,51)(25,39,38,48,40,35,33,49)(30,46,42,52,47,43,45,50);(5,8,9,17)(6,11,12,16)(7,14,15,20)(10,18,13,19)(21,25,27,40)(22,30,31,47)(23,33,26,38)(24,35,37,39)(28,42,41,45)(29,43,44,46)(32,48,36,49)(34,50,51,52);(2,4)(7,15)(10,13)(14,20)(18,19)(25,40)(30,47)(33,38)(35,39)(42,45)(43,46)(48,49)(50,52)Polycyclic presentation:
2^9|a^2=f,b^2=i,c^2=g,d^2=fi,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^d=ei68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6591338);