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 | |
4591 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 40 points:
(2,5)(6,8)(9,10,14,13)(11,15,18,16)(12,17,20,21)(19,22,24,23)(25,26)(27,33)(28,31)(29,32)(30,35)(34,40)(36,38)(37,39);(3,7)(6,8)(9,11)(10,15)(12,19)(13,16)(14,18)(17,22)(20,24)(21,23)(25,26,29,32)(27,30,35,33)(28,31,38,36)(34,39,37,40);(1,2)(3,6)(4,5)(7,8)(11,18)(15,16)(19,24)(22,23)(25,26,29,32)(27,30,35,33)(28,31,38,36)(34,39,37,40);(1,3)(2,6)(4,7)(5,8)(9,11)(10,16)(12,19)(13,15)(14,18)(17,23)(20,24)(21,22)(25,26)(27,30)(28,36)(29,32)(31,38)(33,35)(34,40)(37,39);(1,2)(3,6)(4,5)(7,8)(9,12)(10,17)(11,19)(13,21)(14,20)(15,22)(16,23)(18,24)(25,27)(26,30)(28,37)(29,35)(31,40)(32,33)(34,38)(36,39);(9,13,14,10)(11,16,18,15)(12,17,20,21)(19,22,24,23)(25,28,29,38)(26,31,32,36)(27,34,35,37)(30,39,33,40)Polycyclic presentation:
2^9|a^2=i,b^2=g,c^2=g,f^2=gi,b^a=bg,c^a=ch,d^a=di,e^a=eh,c^b=ci,d^b=dh,d^c=dgi,f^d=fgi,f^e=fgi83 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8653819);