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 | |
4397 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 40 points:
(1,2,5,8)(3,6,4,7)(9,10)(11,16)(12,15)(13,17)(14,19)(18,22)(20,23)(21,24)(25,26,32,31)(27,36,34,29)(28,33,39,37)(30,35,40,38);(1,2)(3,6)(4,7)(5,8)(25,27)(26,29)(28,38)(30,37)(31,36)(32,34)(33,40)(35,39);(1,3,5,4)(2,6,8,7)(9,11)(10,14)(12,18)(13,19)(15,21)(16,17)(20,24)(22,23)(25,28,32,39)(26,33,31,37)(27,35,34,38)(29,30,36,40);(9,12,13,20)(10,15,17,23)(11,18,19,24)(14,21,16,22)(25,29,32,36)(26,34,31,27)(28,30,39,40)(33,38,37,35);(1,4,5,3)(2,7,8,6)(9,11)(10,16)(12,18)(13,19)(14,17)(15,22)(20,24)(21,23)(25,30)(26,35)(27,37)(28,36)(29,39)(31,38)(32,40)(33,34);(9,10)(11,14)(12,15)(13,17)(16,19)(18,21)(20,23)(22,24)(25,31,32,26)(27,36,34,29)(28,37,39,33)(30,35,40,38)Polycyclic presentation:
2^9|a^2=g,c^2=g,d^2=hi,e^2=gi,f^2=i,b^a=bg,c^a=ch,d^a=di,c^b=ci,f^e=fhi98 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7565017);