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 | C24 | C22×D4 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2296 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1687 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)(21,22,26,31)(23,33,36,28)(24,37,39,46)(25,32,27,30)(29,41,44,38)(34,48,50,52)(35,43,40,45)(42,51,47,49);(1,2)(3,4)(5,7,9,15)(6,10,11,13)(8,16,17,14)(12,20,19,18)(21,23,26,36)(22,28,31,33)(24,34,39,50)(25,40,27,35)(29,42,44,47)(30,45,32,43)(37,52,46,48)(38,51,41,49);(2,4)(21,24,25,41,26,39,27,38)(22,29,30,46,31,44,32,37)(23,34,40,49,36,50,35,51)(28,42,45,48,33,47,43,52);(6,11)(10,13)(12,19)(18,20)(21,25,26,27)(22,30,31,32)(23,35,36,40)(24,38,39,41)(28,43,33,45)(29,37,44,46)(34,49,50,51)(42,48,47,52);(2,4)(5,8,9,17)(6,12,11,19)(7,14,15,16)(10,18,13,20)(23,36)(24,39)(28,33)(29,44)(35,40)(37,46)(38,41)(43,45)Polycyclic presentation:
2^9|a^2=i,b^2=f,c^2=gi,d^2=i,e^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^c=ei68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6593141);