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 | |
4352 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 40 points:
(10,14)(15,18)(16,20)(22,24)(25,26)(27,28)(29,37)(30,31)(32,33)(34,38)(35,39)(36,40);(1,2,5,3)(4,6,8,7)(9,10,13,14)(11,15,17,18)(12,16,21,20)(19,22,23,24)(26,31)(27,33)(34,37)(35,40);(2,3)(6,7)(11,17)(15,18)(19,23)(22,24)(25,27,30,33)(26,32,31,28)(29,35,38,40)(34,39,37,36);(2,3)(4,8)(9,10)(11,18)(12,20)(13,14)(15,17)(16,21)(19,22)(23,24)(25,28,30,32)(26,33,31,27)(29,36,38,39)(34,40,37,35);(1,3,5,2)(4,7,8,6)(9,11)(10,15)(12,19)(13,17)(14,18)(16,22)(20,24)(21,23)(25,26)(27,32)(28,33)(29,34)(30,31)(35,39)(36,40)(37,38);(1,4)(2,6)(3,7)(5,8)(9,12)(10,16)(11,19)(13,21)(14,20)(15,22)(17,23)(18,24)(25,29)(26,34)(27,35)(28,36)(30,38)(31,37)(32,39)(33,40)Polycyclic presentation:
2^9|b^2=gi,c^2=h,d^2=h,e^2=hi,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,d^b=dgi,e^b=eh,e^c=egi,e^d=egi,f^d=fgi83 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 9012487);