Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6470894 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C27 |
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 ◁ #2295 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1180 conjugacy classes of subgroups.
| minimal degree | μ(G) = 56 |
| transitive degree | μt(G) = 512 |
A minimal faithful permutation representation — 5 generators on 56 points:
(1,2,5,8)(3,7,4,6)(9,10)(11,14)(12,15)(13,16)(17,21)(18,22)(19,23)(20,24)(25,26)(27,39)(28,45)(29,35)(30,38)(31,37)(32,36)(33,42)(34,47)(40,50)(41,54)(43,52)(44,53)(46,51)(48,56)(49,55);(1,3,5,4)(2,6,8,7)(9,11)(10,14)(12,19)(13,17)(15,23)(16,21)(18,20)(22,24)(25,27,31,43,32,44,30,42)(26,33,38,53,36,52,37,39)(28,29,47,49,46,40,41,48)(34,35,45,56,54,50,51,55);(1,3,5,4)(2,7,8,6)(11,17)(14,21)(18,19)(22,23)(25,28,31,47,32,46,30,41)(26,34,38,45,36,54,37,51)(27,40,43,48,44,29,42,49)(33,50,53,55,52,35,39,56);(9,12,13,20)(10,15,16,24)(11,18,17,19)(14,22,21,23)(25,29,32,40)(26,35,36,50)(27,41,44,47)(28,42,46,43)(30,48,31,49)(33,51,52,45)(34,39,54,53)(37,55,38,56);(1,4,5,3)(2,6,8,7)(10,16)(14,21)(15,24)(22,23)(26,36)(28,46)(29,40)(33,52)(37,38)(39,53)(41,47)(48,49)Polycyclic presentation:
2^9|a^2=fg,b^2=fi,c^2=fi,d^2=h,e^2=fg,f^2=i,g^2=i,b^a=bf,c^a=cg,e^a=eh,f^a=fi,g^a=gi,c^b=ch,d^b=dh,e^c=ei,e^d=ei80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6393785);