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 | |
4555 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 40 points:
(9,10,14,18)(11,12,16,15)(13,17,21,24)(19,20,23,22)(25,26)(27,34)(28,33)(29,30)(31,32)(35,40)(36,37)(38,39);(1,2,5,6)(3,7,8,4)(9,11,14,16)(10,15,18,12)(13,19,21,23)(17,22,24,20)(26,32)(29,40)(33,34)(36,38);(2,6)(4,7)(13,21)(17,24)(19,23)(20,22)(25,27,31,28)(26,33,32,34)(29,36,40,38)(30,39,35,37);(2,6)(3,8)(9,10)(11,12)(13,17)(14,18)(15,16)(19,20)(21,24)(22,23)(25,28,31,27)(26,34,32,33)(29,36,40,38)(30,39,35,37);(1,3)(2,4)(5,8)(6,7)(9,12)(10,16)(11,18)(13,20)(14,15)(17,23)(19,24)(21,22)(25,29)(26,35)(27,36)(28,38)(30,32)(31,40)(33,37)(34,39);(1,4)(2,3)(5,7)(6,8)(9,13,14,21)(10,17,18,24)(11,19,16,23)(12,20,15,22)(25,30,31,35)(26,29,32,40)(27,37,28,39)(33,38,34,36)Polycyclic presentation:
2^9|a^2=gh,b^2=ghi,c^2=h,d^2=h,f^2=g,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dghi,e^b=egi,f^c=fgi,e^d=egi92 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8295322);