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 | |
1728 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)(21,22,25,29)(23,31,33,27)(24,35,37,51)(26,39,36,30)(28,42,43,38)(32,46,48,50)(34,41,47,45)(40,49,44,52);(1,2)(3,4)(5,7,9,15)(6,10,11,13)(8,16,17,14)(12,20,19,18)(21,23,25,33)(22,27,29,31)(24,32,37,48)(26,34,36,47)(28,40,43,44)(30,41,39,45)(35,50,51,46)(38,49,42,52);(2,4)(21,24,36,42,25,37,26,38)(22,28,39,51,29,43,30,35)(23,32,47,52,33,48,34,49)(27,40,45,46,31,44,41,50);(6,11)(10,13)(12,19)(18,20)(22,29)(23,33)(24,37)(30,39)(34,47)(38,42)(40,44)(46,50);(2,4)(5,8,9,17)(6,12,11,19)(7,14,15,16)(10,18,13,20)(23,33)(24,37)(27,31)(28,43)(34,47)(35,51)(38,42)(41,45)Polycyclic presentation:
2^9|a^2=i,b^2=f,c^2=gi,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, 6593137);