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 | |
4595 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)(19,20)(23,24)(25,26)(27,30)(28,33)(29,32)(31,36)(34,37)(35,39)(38,40);(3,7)(6,8)(10,15)(12,22)(16,18)(19,24)(26,27)(31,33)(32,34)(38,39);(1,2,5,4)(3,6,7,8)(25,26,30,27)(28,31,36,33)(29,32,37,34)(35,38,40,39);(1,3)(2,6)(4,8)(5,7)(9,11,14,21)(10,16,15,18)(12,19,22,24)(13,23,17,20)(25,27,30,26)(28,33,36,31)(29,34,37,32)(35,39,40,38);(1,4,5,2)(3,8,7,6)(9,12)(10,17)(11,19)(13,15)(14,22)(16,20)(18,23)(21,24)(25,28,30,36)(26,31,27,33)(29,35,37,40)(32,38,34,39);(9,13,14,17)(10,12,15,22)(11,20,21,23)(16,24,18,19)(25,29,30,37)(26,32,27,34)(28,35,36,40)(31,38,33,39)Polycyclic presentation:
2^9|c^2=h,d^2=g,e^2=h,f^2=g,b^a=bg,c^a=ch,d^a=dg,e^a=eghi,c^b=ci,d^b=dh,e^b=egi,f^d=fgi83 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8379691);