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 | |
4455 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)(3,7)(4,6)(5,8)(9,10,15,17)(11,16,21,23)(12,18,22,24)(13,19,20,14)(25,26,30,32)(27,33,34,29)(28,31,38,40)(35,39,36,37);(1,2,4,6)(3,5,8,7)(9,11)(10,16)(12,20)(13,22)(14,18)(15,21)(17,23)(19,24)(25,26)(27,29)(28,31)(30,32)(33,34)(35,37)(36,39)(38,40);(1,2)(3,5)(4,6)(7,8)(10,17)(11,21)(12,22)(14,19)(27,34)(29,33)(35,36)(37,39);(1,3)(2,5)(4,8)(6,7)(9,12,15,22)(10,18,17,24)(11,20,21,13)(14,23,19,16)(25,27)(26,29)(28,36)(30,34)(31,39)(32,33)(35,38)(37,40);(9,13,15,20)(10,19,17,14)(11,12,21,22)(16,18,23,24)(25,28,30,38)(26,31,32,40)(27,35,34,36)(29,37,33,39);(9,14,15,19)(10,20,17,13)(11,18,21,24)(12,23,22,16)(25,29)(26,27)(28,37)(30,33)(31,35)(32,34)(36,40)(38,39)Polycyclic presentation:
2^9|a^2=gi,b^2=hi,d^2=h,e^2=gi,f^2=h,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,d^c=dgi,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, 8827936);