Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6994139 |
| 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 ◁ #2396 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1317 conjugacy classes of subgroups.
| minimal degree | μ(G) = 56 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 56 points:
(1,2,5,4)(3,7,8,6)(12,20)(15,21)(19,24)(22,23)(25,26,31,33,30,35,29,36)(27,32,42,44,39,46,38,47)(28,34,43,48,40,50,41,49)(37,45,53,54,51,56,52,55);(1,3)(2,6)(4,7)(5,8)(9,10,13,16)(11,17,18,14)(12,21,20,15)(19,22,24,23)(25,27,30,39)(26,32,35,46)(28,37,40,51)(29,38,31,42)(33,44,36,47)(34,45,50,56)(41,52,43,53)(48,54,49,55);(1,4,5,2)(3,7,8,6)(9,11)(10,14)(12,19)(13,18)(15,22)(16,17)(20,24)(21,23)(26,33)(29,31)(32,44)(34,48)(35,36)(38,42)(41,43)(45,54)(46,47)(49,50)(52,53)(55,56);(3,8)(6,7)(11,18)(14,17)(19,24)(22,23)(25,28,30,40)(26,34,35,50)(27,37,39,51)(29,41,31,43)(32,45,46,56)(33,48,36,49)(38,52,42,53)(44,54,47,55);(9,12,13,20)(10,15,16,21)(11,19,18,24)(14,22,17,23)(26,35)(28,40)(32,46)(33,36)(37,51)(41,43)(44,47)(52,53)Polycyclic presentation:
2^9|a^2=fgi,b^2=hi,c^2=f,d^2=i,e^2=h,g^2=i,b^a=bf,c^a=cg,e^a=ehi,c^b=ch,d^b=df,e^b=eh,d^c=dh,g^c=gi,e^d=ei71 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7090271);