Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6470355 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C27 |
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 ◁ C4×#117 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1232 conjugacy classes of subgroups.
| minimal degree | μ(G) = 56 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 56 points:
(1,2,4,6)(3,7,8,5)(9,10,13,15)(11,14,17,21)(12,19,20,16)(18,24,23,22)(25,26)(27,35)(28,32)(29,42)(30,33)(31,38)(34,43)(36,49)(37,50)(39,54)(40,47)(41,56)(44,52)(45,53)(46,55)(48,51);(9,11)(10,14)(12,18)(13,17)(15,21)(16,22)(19,24)(20,23)(25,27,29,38,30,37,43,53)(26,31,34,35,33,45,42,50)(28,39,41,55,40,36,51,52)(32,46,48,54,47,44,56,49);(1,3,4,8)(2,5,6,7)(10,15)(11,17)(16,19)(18,23)(25,28)(26,32)(27,36)(29,41)(30,40)(31,44)(33,47)(34,48)(35,49)(37,39)(38,52)(42,56)(43,51)(45,46)(50,54)(53,55);(11,17)(14,21)(18,23)(22,24)(27,37)(28,40)(31,45)(32,47)(35,50)(38,53)(41,51)(48,56);(9,12,13,20)(10,16,15,19)(11,18,17,23)(14,22,21,24)(26,33)(31,45)(32,47)(34,42)(35,50)(44,46)(48,56)(49,54)Polycyclic presentation:
2^9|a^2=g,b^2=f,c^2=ghi,e^2=hi,f^2=i,b^a=bf,c^a=cg,e^a=eh,f^a=fi,c^b=ch,d^b=dh,d^c=di80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6406543);