Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6667556 |
| 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 | C22×D4 | C24 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2399 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1655 conjugacy classes of subgroups.
| minimal degree | μ(G) = 52 |
| transitive degree | μt(G) = 256 |
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)(23,32)(24,37)(25,30)(26,31)(27,29)(28,36)(33,44)(34,42)(35,49)(38,48)(39,52)(40,47)(41,45)(43,51)(46,50);(5,7)(6,10)(8,14)(9,15)(11,18)(12,13)(16,19)(17,20)(21,23)(22,28)(24,34)(25,39)(26,36)(27,33)(29,42)(30,46)(31,32)(35,41)(37,44)(38,51)(40,50)(43,45)(47,52)(48,49);(1,2,3,4)(23,33)(24,27)(28,42)(29,37)(32,44)(34,36)(35,50)(38,41)(39,51)(43,52)(45,48)(46,49);(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)(25,38,40,41)(28,42,32,44)(30,45,47,48)(35,39,51,50)(43,52,49,46);(7,15)(10,13)(14,20)(18,19)(21,25,26,40)(22,30,31,47)(23,35,36,51)(24,38,27,41)(28,43,32,49)(29,45,37,48)(33,50,34,39)(42,52,44,46)Polycyclic presentation:
2^9|a^2=fi,c^2=g,d^2=f,e^2=i,h^2=i,b^a=bf,c^a=cg,d^a=df,c^b=ch,e^b=efh,h^b=hi,d^c=di,h^c=hi,h^e=hi74 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6591113);