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 | |
4720 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 40 points:
(2,4)(6,8)(9,10,14,13)(11,17,18,15)(12,20,21,16)(19,22,24,23)(25,26)(27,33)(28,36)(29,30)(31,35)(32,37)(34,38)(39,40);(3,7)(6,8)(10,13)(12,21)(15,17)(19,24)(25,26)(27,31)(28,36)(29,30)(32,37)(33,35)(34,39)(38,40);(1,2,5,4)(3,6,7,8)(11,18)(15,17)(19,24)(22,23)(26,30)(31,33)(32,36)(38,39);(9,11,14,18)(10,15,13,17)(12,19,21,24)(16,22,20,23)(25,26)(27,31)(28,32)(29,30)(33,35)(34,38)(36,37)(39,40);(1,3)(2,6)(4,8)(5,7)(9,12,14,21)(10,16,13,20)(11,19,18,24)(15,22,17,23)(25,27,29,35)(26,31,30,33)(28,34,37,40)(32,38,36,39);(1,4,5,2)(3,8,7,6)(9,13,14,10)(11,17,18,15)(12,20,21,16)(19,23,24,22)(25,28,29,37)(26,32,30,36)(27,34,35,40)(31,38,33,39)Polycyclic presentation:
2^9|a^2=g,c^2=hi,d^2=g,e^2=gi,f^2=gh,b^a=bg,c^a=ch,d^a=dg,e^a=egi,f^a=fh,c^b=ci,e^b=eghi,f^b=fgi,d^c=dgi80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8664275);