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 | |
4550 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)(3,7)(4,8)(5,6)(9,10,15,20)(11,17,12,16)(13,14,19,18)(21,22,23,24)(28,36)(32,37)(34,40)(38,39);(2,6)(3,8)(9,11,15,12)(10,16,20,17)(13,23,19,21)(14,22,18,24)(25,26)(27,31)(28,37)(29,33)(30,35)(32,36)(34,38)(39,40);(1,3)(2,4)(5,8)(6,7)(9,11,15,12)(10,17,20,16)(13,21,19,23)(14,22,18,24)(25,27)(26,30)(28,34)(29,35)(31,33)(32,38)(36,40)(37,39);(1,3)(2,4)(5,8)(6,7)(9,12,15,11)(10,17,20,16)(13,23,19,21)(14,22,18,24)(25,27)(26,31)(28,34)(29,35)(30,33)(32,39)(36,40)(37,38);(1,4)(2,3)(5,7)(6,8)(9,13,15,19)(10,18,20,14)(11,21,12,23)(16,22,17,24)(25,28)(26,32)(27,34)(29,36)(30,38)(31,39)(33,37)(35,40);(9,14,15,18)(10,19,20,13)(11,22,12,24)(16,21,17,23)(25,26)(27,30)(28,32)(29,33)(31,35)(34,38)(36,37)(39,40)Polycyclic presentation:
2^9|a^2=gh,b^2=gh,c^2=gh,d^2=gh,e^2=gh,f^2=gh,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, 8382380);