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 | |
4591 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)(3,4)(5,6)(7,8)(9,10,14,13)(11,17,18,15)(12,16,20,21)(19,23,24,22)(25,26,30,35)(27,31,38,32)(28,33,34,29)(36,40,39,37);(2,6)(4,7)(10,13)(15,17)(16,21)(22,23)(25,27)(26,31)(28,39)(29,40)(30,38)(32,35)(33,37)(34,36);(2,6)(3,8)(11,18)(12,20)(15,17)(16,21)(25,27,30,38)(26,32,35,31)(28,39,34,36)(29,37,33,40);(1,3,5,8)(2,7,6,4)(9,11)(10,15)(12,19)(13,17)(14,18)(16,22)(20,24)(21,23)(25,27)(26,31)(28,36)(29,37)(30,38)(32,35)(33,40)(34,39);(1,2)(3,7)(4,8)(5,6)(9,12,14,20)(10,16,13,21)(11,19,18,24)(15,22,17,23)(25,28,30,34)(26,33,35,29)(27,36,38,39)(31,40,32,37);(1,4,5,7)(2,8,6,3)(9,13,14,10)(11,17,18,15)(12,21,20,16)(19,23,24,22)(25,29,30,33)(26,34,35,28)(27,37,38,40)(31,39,32,36)Polycyclic presentation:
2^9|a^2=gh,c^2=i,d^2=hi,e^2=gh,f^2=gi,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,f^b=fgi,d^c=dgi,e^c=egi86 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8725414);