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 | |
4607 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 40 points:
(4,8)(6,7)(9,10,14,16)(11,17,18,12)(13,21,22,15)(19,20,24,23)(25,26,31,35)(27,32,36,40)(28,30,38,33)(29,39,37,34);(1,2,5,3)(4,6,8,7)(9,10)(11,12)(13,15)(14,16)(17,18)(19,20)(21,22)(23,24)(25,27,31,36)(26,32,35,40)(28,37,38,29)(30,34,33,39);(2,3)(6,7)(11,18)(12,17)(19,24)(20,23)(25,28,31,38)(26,33,35,30)(27,29,36,37)(32,34,40,39);(1,2,5,3)(4,6,8,7)(9,11)(10,12)(13,19)(14,18)(15,20)(16,17)(21,23)(22,24)(25,28)(26,30)(27,29)(31,38)(32,39)(33,35)(34,40)(36,37);(1,3,5,2)(4,7,8,6)(9,12)(10,11)(13,20)(14,17)(15,19)(16,18)(21,24)(22,23)(25,29)(26,34)(27,28)(30,40)(31,37)(32,33)(35,39)(36,38);(1,4)(2,6)(3,7)(5,8)(9,13,14,22)(10,15,16,21)(11,19,18,24)(12,20,17,23)(25,30)(26,28)(27,34)(29,40)(31,33)(32,37)(35,38)(36,39)Polycyclic presentation:
2^9|a^2=gh,b^2=i,c^2=h,d^2=hi,e^2=hi,f^2=g,b^a=bg,c^a=ch,d^a=dg,e^a=eh,f^a=fgi,c^b=ci,d^b=dh,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, 8038720);