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 | |
4422 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 40 points:
(9,10)(11,15)(12,20)(13,14)(16,21)(17,19)(18,23)(22,24)(25,26,32,36)(27,34,28,33)(29,35,30,31)(37,40,38,39);(1,2)(3,4)(5,6)(7,8)(9,10)(11,17)(12,20)(13,14)(15,19)(16,21)(18,22)(23,24)(25,27,32,28)(26,33,36,34)(29,38,30,37)(31,39,35,40);(2,6)(4,7)(10,14)(15,17)(16,20)(22,23)(25,28,32,27)(26,34,36,33)(29,38,30,37)(31,39,35,40);(9,11)(10,15)(12,18)(13,19)(14,17)(16,22)(20,23)(21,24)(25,29,32,30)(26,31,36,35)(27,37,28,38)(33,40,34,39);(1,3)(2,7)(4,6)(5,8)(9,12)(10,16)(11,18)(13,21)(14,20)(15,22)(17,23)(19,24)(25,30,32,29)(26,35,36,31)(27,38,28,37)(33,39,34,40);(1,4,5,7)(2,3,6,8)(9,10)(11,15)(12,16)(13,14)(17,19)(18,22)(20,21)(23,24)(25,31,32,35)(26,29,36,30)(27,39,28,40)(33,38,34,37)Polycyclic presentation:
2^9|a^2=g,b^2=g,c^2=g,d^2=g,e^2=g,f^2=hi,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,e^b=egi,f^c=fgi86 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8139472);