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 | |
4564 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 40 points:
(1,2,5,8)(3,6,4,7)(9,10,14,16)(11,17,18,12)(13,15,22,24)(19,23,21,20)(25,26,31,33)(27,37,38,32)(28,40,39,34)(29,30,36,35);(1,2)(3,6)(4,7)(5,8)(9,11)(10,12)(13,21)(14,18)(15,23)(16,17)(19,22)(20,24)(25,27)(26,32)(28,36)(29,39)(30,40)(31,38)(33,37)(34,35);(11,18)(12,17)(19,21)(20,23)(26,33)(27,38)(29,36)(34,40);(1,3,5,4)(2,6,8,7)(9,11)(10,12)(13,19)(14,18)(15,20)(16,17)(21,22)(23,24)(25,28,31,39)(26,34,33,40)(27,36,38,29)(30,32,35,37);(9,12)(10,11)(13,20)(14,17)(15,19)(16,18)(21,24)(22,23)(25,29,31,36)(26,35,33,30)(27,39,38,28)(32,34,37,40);(1,4,5,3)(2,7,8,6)(9,13,14,22)(10,15,16,24)(11,19,18,21)(12,20,17,23)(25,30,31,35)(26,36,33,29)(27,34,38,40)(28,32,39,37)Polycyclic presentation:
2^9|a^2=g,d^2=ghi,e^2=h,f^2=g,b^a=bg,c^a=ch,d^a=di,e^a=eh,c^b=ci,f^b=fi,d^c=dhi,e^c=ei92 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7648703);