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 | |
4512 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 40 points:
(2,4)(6,8)(9,10)(11,18)(12,13)(14,15)(16,21)(17,22)(19,24)(20,23)(25,26)(27,35)(28,31)(29,33)(30,40)(32,36)(34,37)(38,39);(3,7)(6,8)(10,15)(12,17)(16,18)(19,23)(26,28)(27,36)(29,38)(34,40);(1,2,5,4)(3,6,7,8)(25,27)(26,32)(28,35)(29,37)(30,38)(31,36)(33,40)(34,39);(1,3)(2,6)(4,8)(5,7)(9,11,14,21)(10,16,15,18)(12,19,17,23)(13,20,22,24)(25,28,31,26)(27,35,36,32)(29,39,38,33)(30,40,37,34);(1,3)(2,6)(4,8)(5,7)(9,12,14,17)(10,13,15,22)(11,19,21,23)(16,20,18,24)(25,29,31,38)(26,33,28,39)(27,37,36,30)(32,40,35,34);(1,4,5,2)(3,8,7,6)(9,13)(10,17)(11,20)(12,15)(14,22)(16,23)(18,19)(21,24)(25,30,31,37)(26,34,28,40)(27,38,36,29)(32,39,35,33)Polycyclic presentation:
2^9|c^2=hi,d^2=g,e^2=g,f^2=h,b^a=bg,c^a=ch,d^a=dg,f^a=fghi,c^b=ci,d^b=dh,e^b=eghi92 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8000454);