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 | |
4420 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 40 points:
(1,2)(3,7)(4,6)(5,8)(25,26,30,34)(27,35,36,31)(28,32,38,39)(29,33,37,40);(1,2,4,6)(3,5,8,7)(9,10,14,18)(11,16,21,15)(12,13,17,24)(19,23,22,20)(27,36)(29,37)(31,35)(33,40);(1,2)(3,5)(4,6)(7,8)(9,11,14,21)(10,15,18,16)(12,22,17,19)(13,23,24,20)(25,27,30,36)(26,31,34,35)(28,29,38,37)(32,40,39,33);(1,3)(2,5)(4,8)(6,7)(9,12,14,17)(10,13,18,24)(11,19,21,22)(15,20,16,23)(25,28,30,38)(26,32,34,39)(27,37,36,29)(31,33,35,40);(9,11)(10,16)(12,22)(13,20)(14,21)(15,18)(17,19)(23,24)(25,29,30,37)(26,33,34,40)(27,28,36,38)(31,39,35,32);(9,13)(10,17)(11,20)(12,18)(14,24)(15,22)(16,19)(21,23)(25,26)(27,35)(28,32)(29,33)(30,34)(31,36)(37,40)(38,39)Polycyclic presentation:
2^9|a^2=h,b^2=hi,c^2=gi,d^2=gi,e^2=h,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,d^c=dgi,e^c=egi,f^c=fgi,e^d=egi86 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8901496);