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 | C2×#466 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2296 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1739 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)(23,32)(24,36)(25,31)(26,28)(27,42)(29,35)(30,39)(33,45)(34,50)(37,49)(38,48)(40,43)(41,47)(44,51)(46,52);(1,2)(3,4)(5,7,9,15)(6,10,11,13)(8,16,17,14)(12,20,19,18)(22,28)(23,33)(30,36)(31,42)(34,37)(35,40)(47,48)(51,52);(2,4)(21,23,27,35,26,33,25,40)(22,29,31,32,28,43,42,45)(24,37,38,51,39,34,41,52)(30,46,48,50,36,44,47,49);(6,11)(10,13)(12,19)(18,20)(21,24)(22,30)(23,34)(25,41)(26,39)(27,38)(28,36)(29,44)(31,48)(32,49)(33,37)(35,52)(40,51)(42,47)(43,46)(45,50);(2,4)(5,8,9,17)(6,12,11,19)(7,14,15,16)(10,18,13,20)(21,25,26,27)(22,31,28,42)(23,35,33,40)(24,38,39,41)(29,45,43,32)(30,47,36,48)(34,51,37,52)(44,49,46,50)Polycyclic presentation:
2^9|b^2=fi,c^2=g,e^2=f,g^2=i,b^a=bf,c^a=cg,d^a=df,g^a=gi,c^b=ch,e^b=efh,d^c=di,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, 6593250);