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 ◁ #2285 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1715 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)(21,22,26,28)(23,32,33,49)(24,37,38,30)(25,31,27,42)(29,35,43,36)(34,50,39,48)(40,47,41,46)(44,51,45,52);(1,2)(3,4)(5,7,9,15)(6,10,11,13)(8,16,17,14)(12,20,19,18)(22,28)(23,33)(24,38)(31,42)(35,36)(40,41)(44,45)(48,50);(2,4)(21,23,27,36,26,33,25,35)(22,29,31,32,28,43,42,49)(24,39,40,52,38,34,41,51)(30,45,47,50,37,44,46,48);(6,11)(10,13)(12,19)(18,20)(21,24,26,38)(22,30,28,37)(23,34,33,39)(25,41,27,40)(29,44,43,45)(31,47,42,46)(32,48,49,50)(35,52,36,51);(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)(23,35,33,36)(24,40,38,41)(29,32,43,49)(30,46,37,47)(34,51,39,52)(44,50,45,48)Polycyclic presentation:
2^9|a^2=i,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^d=ei68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6593211);