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)(25,26,31,33,29,35,30,36)(27,37,44,48,39,32,40,46)(28,34,45,49,42,51,43,52)(38,50,54,56,41,47,53,55);(1,3)(2,6)(4,7)(5,8)(9,10,13,16)(11,17,18,14)(12,21,20,15)(19,22,24,23)(25,27,29,39)(26,32,35,37)(28,41,42,38)(30,40,31,44)(33,46,36,48)(34,50,51,47)(43,54,45,53)(49,56,52,55);(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)(30,31)(32,46)(34,49)(35,36)(37,48)(40,44)(43,45)(47,55)(50,56)(51,52)(53,54);(3,8)(6,7)(11,18)(14,17)(19,24)(22,23)(25,28,29,42)(26,34,35,51)(27,38,39,41)(30,43,31,45)(32,47,37,50)(33,49,36,52)(40,53,44,54)(46,55,48,56);(9,12,13,20)(10,15,16,21)(11,19,18,24)(14,22,17,23)(26,35)(28,42)(32,37)(33,36)(38,41)(43,45)(46,48)(53,54)Polycyclic presentation:
2^9|a^2=fg,b^2=hi,c^2=fi,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, 7090259);