Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6667604 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C28 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle, prime radical | C23 | C23×D4 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2399 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1618 conjugacy classes of subgroups.
| minimal degree | μ(G) = 52 |
| transitive degree | μt(G) = 512 |
A minimal faithful permutation representation — 5 generators on 52 points:
(2,4)(5,6,9,12)(7,13,15,10)(8,16,17,11)(14,18,20,19)(21,22,26,31)(23,32,36,28)(24,37,27,29)(25,39,41,30)(33,45,34,43)(35,44,51,49)(38,46,42,48)(40,47,50,52);(5,7)(6,10)(8,14)(9,15)(11,18)(12,13)(16,19)(17,20)(21,23,26,36)(22,28,31,32)(24,34,27,33)(25,40,41,50)(29,43,37,45)(30,47,39,52)(35,38,51,42)(44,48,49,46);(1,2,3,4)(23,33)(24,27)(28,43)(29,37)(32,45)(34,36)(35,50)(38,42)(40,51)(44,52)(46,48)(47,49);(5,8,9,17)(6,11,12,16)(7,14,15,20)(10,18,13,19)(21,24,26,27)(22,29,31,37)(23,34,36,33)(25,38,41,42)(28,43,32,45)(30,46,39,48)(35,40,51,50)(44,52,49,47);(7,15)(10,13)(14,20)(18,19)(21,25,26,41)(22,30,31,39)(23,35,36,51)(24,38,27,42)(28,44,32,49)(29,46,37,48)(33,50,34,40)(43,52,45,47)Polycyclic presentation:
2^9|a^2=f,b^2=i,c^2=g,d^2=f,e^2=i,h^2=i,b^a=bf,c^a=cg,d^a=df,e^a=ei,c^b=ch,e^b=efh,h^b=hi,d^c=di,h^c=hi,h^e=hi62 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6591131);