Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| automorphism | ? |
| homology | Abelianization (H1) = C26 · Schur multiplier (H2) = C212 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, commutator, Frattini, socle, cosocle | C23 | C26 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
4682 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 40 points:
(1,2,5,7)(3,8,4,6)(9,10)(11,18)(12,16)(13,21)(14,15)(17,23)(19,20)(22,24)(25,26,29,30)(27,31,35,33)(28,36,37,32)(34,39,40,38);(3,4)(6,8)(10,15)(12,18)(17,21)(20,22)(25,26)(27,33)(28,32)(29,30)(31,35)(34,39)(36,37)(38,40);(1,3,5,4)(2,6,7,8)(10,15)(11,16)(17,21)(19,24)(26,30)(31,33)(32,36)(38,39);(3,4)(6,8)(9,11)(10,12)(13,19)(14,16)(15,18)(17,20)(21,22)(23,24)(25,26)(27,31)(28,32)(29,30)(33,35)(34,38)(36,37)(39,40);(1,4,5,3)(2,6,7,8)(9,12)(10,16)(11,15)(13,22)(14,18)(17,19)(20,23)(21,24)(25,27,29,35)(26,31,30,33)(28,34,37,40)(32,38,36,39);(1,2)(3,6)(4,8)(5,7)(9,13,14,23)(10,17,15,21)(11,19,16,24)(12,20,18,22)(25,28,29,37)(26,32,30,36)(27,34,35,40)(31,38,33,39)Polycyclic presentation:
2^9|a^2=i,c^2=gh,e^2=i,f^2=g,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,d^c=dh,e^d=egi,f^e=fgi86 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 10054233);