Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6469995 |
| 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 | C2×#121 | C23 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C4×#117 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1392 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)(3,8)(4,6)(5,7)(9,10)(11,14)(12,15)(13,16)(17,21)(18,22)(19,23)(20,24)(25,26,30,33)(27,35,36,50)(28,32,40,48)(29,42,44,34)(31,43,45,38)(37,51,39,53)(41,56,55,49)(46,54,47,52);(9,11)(10,14)(12,19)(13,18)(15,23)(16,22)(17,20)(21,24)(25,27)(26,31)(28,39)(29,43)(30,36)(32,47)(33,45)(34,50)(35,42)(37,40)(38,44)(41,52)(46,48)(49,51)(53,56)(54,55);(1,3,5,4)(2,6,7,8)(9,12,13,20)(10,15,16,24)(11,17,18,19)(14,21,22,23)(27,36)(28,40)(31,45)(32,48)(35,50)(38,43)(41,55)(49,56);(11,18)(14,22)(17,19)(21,23)(25,28)(26,32)(27,37)(29,41)(30,40)(31,46)(33,48)(34,49)(35,51)(36,39)(38,52)(42,56)(43,54)(44,55)(45,47)(50,53);(1,4,5,3)(2,6,7,8)(10,16)(14,22)(15,24)(21,23)(26,33)(28,40)(31,45)(34,42)(35,50)(37,39)(41,55)(52,54)Polycyclic presentation:
2^9|a^2=i,c^2=ghi,e^2=g,f^2=i,b^a=bf,c^a=cg,e^a=eh,f^a=fi,c^b=ch,d^b=dh,f^b=fi,d^c=di,e^d=ei74 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6388178);