Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6499792 |
| 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 ◁ #2285 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1324 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)(9,10,13,16)(11,14,17,20)(12,15,19,22)(18,21,23,24)(25,26)(27,36)(28,41)(29,34)(30,46)(31,35)(32,39)(33,43)(37,54)(38,48)(40,52)(42,53)(44,51)(45,50)(47,56)(49,55);(9,11)(10,14)(12,18)(13,17)(15,21)(16,20)(19,23)(22,24)(25,27,31,40)(26,32,35,48)(28,42,43,37)(29,39,30,38)(33,49,41,47)(34,52,46,36)(44,55,45,56)(50,54,51,53);(1,3,4,8)(2,5,6,7)(11,17)(12,19)(14,20)(15,22)(25,28,31,43)(26,33,35,41)(27,37,40,42)(29,45,30,44)(32,47,48,49)(34,51,46,50)(36,53,52,54)(38,56,39,55);(9,12,13,19)(10,15,16,22)(11,18,17,23)(14,21,20,24)(28,43)(33,41)(37,42)(44,45)(47,49)(50,51)(53,54)(55,56);(11,17)(14,20)(18,23)(21,24)(25,29,31,30)(26,34,35,46)(27,38,40,39)(28,44,43,45)(32,36,48,52)(33,50,41,51)(37,55,42,56)(47,54,49,53)Polycyclic presentation:
2^9|a^2=gh,b^2=i,c^2=g,d^2=hi,e^2=i,f^2=i,b^a=bf,c^a=cg,f^a=fi,c^b=ch,e^b=eh,f^b=fi,d^c=dh,e^c=ei74 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6435556);