Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6500416 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C26×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C23×D4 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| prime radical | C2×#121 | C23 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2285 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1359 conjugacy classes of subgroups.
| minimal degree | μ(G) = 56 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 56 points:
(1,2,4,5)(3,7,8,6)(9,10,13,16)(11,14,17,20)(12,15,19,22)(18,21,23,24)(25,26)(27,36)(28,33)(29,34)(30,45)(31,35)(32,38)(37,53)(39,47)(40,56)(41,50)(42,51)(43,48)(44,52)(46,55)(49,54);(2,5)(6,7)(9,11)(10,14)(12,18)(13,17)(15,21)(16,20)(19,23)(22,24)(25,27,30,38,31,39,29,43)(26,32,34,36,35,48,45,47)(28,40,42,54,41,37,44,55)(33,49,52,56,50,46,51,53);(1,3,4,8)(2,6,5,7)(11,17)(12,19)(14,20)(15,22)(25,28)(26,33)(27,37)(29,44)(30,42)(31,41)(32,46)(34,52)(35,50)(36,53)(38,55)(39,40)(43,54)(45,51)(47,56)(48,49);(9,12,13,19)(10,15,16,22)(11,18,17,23)(14,21,20,24)(28,41)(33,50)(37,40)(42,44)(46,49)(51,52)(53,56)(54,55);(11,17)(14,20)(18,23)(21,24)(25,29,31,30)(26,34,35,45)(27,38,39,43)(28,42,41,44)(32,47,48,36)(33,51,50,52)(37,54,40,55)(46,53,49,56)Polycyclic presentation:
2^9|a^2=ghi,b^2=fg,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,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, 6435800);