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 | |
4351 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)(4,5)(6,8)(9,10)(11,15)(12,18)(13,16)(14,17)(19,23)(20,24)(21,22)(25,26,30,35)(27,32,38,31)(28,33,34,29)(36,40,39,37);(2,5)(6,7)(11,12)(15,18)(19,20)(23,24)(25,27,30,38)(26,31,35,32)(28,39,34,36)(29,37,33,40);(1,2)(3,6)(4,5)(7,8)(9,11,14,12)(10,15,17,18)(13,19,22,20)(16,23,21,24)(25,27,30,38)(26,31,35,32)(28,39,34,36)(29,37,33,40);(1,3,4,8)(2,6,5,7)(9,12,14,11)(10,15,17,18)(13,20,22,19)(16,23,21,24)(25,27)(26,32)(28,36)(29,37)(30,38)(31,35)(33,40)(34,39);(9,10)(11,18)(12,15)(13,21)(14,17)(16,22)(19,23)(20,24)(25,28)(26,33)(27,36)(29,35)(30,34)(31,37)(32,40)(38,39);(9,13)(10,16)(11,19)(12,20)(14,22)(15,23)(17,21)(18,24)(25,29,30,33)(26,34,35,28)(27,37,38,40)(31,36,32,39)Polycyclic presentation:
2^9|a^2=h,b^2=h,c^2=gi,d^2=i,f^2=h,b^a=bg,c^a=ch,d^a=di,f^a=fh,c^b=ci,d^b=dgi,e^b=eh,d^c=dh,e^c=egi,f^e=fgi83 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 9954474);