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 | |
4581 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 40 points:
(1,2,5,6)(3,4,7,8)(9,10,15,20)(11,17,12,16)(13,14,19,18)(21,22,23,24)(25,26,31,30)(27,32,37,36)(28,38,29,33)(34,40,35,39);(2,6)(3,7)(9,11,15,12)(10,16,20,17)(13,23,19,21)(14,22,18,24)(25,27,31,37)(26,32,30,36)(28,35,29,34)(33,40,38,39);(1,3)(2,4)(5,7)(6,8)(9,11,15,12)(10,17,20,16)(13,21,19,23)(14,22,18,24)(25,28,31,29)(26,33,30,38)(27,34,37,35)(32,39,36,40);(1,4)(2,3)(5,8)(6,7)(9,12,15,11)(10,17,20,16)(13,23,19,21)(14,22,18,24)(25,29,31,28)(26,33,30,38)(27,35,37,34)(32,39,36,40);(9,13,15,19)(10,18,20,14)(11,21,12,23)(16,22,17,24)(25,27)(26,32)(28,34)(29,35)(30,36)(31,37)(33,39)(38,40);(9,14,15,18)(10,19,20,13)(11,22,12,24)(16,21,17,23)(25,30,31,26)(27,36,37,32)(28,38,29,33)(34,40,35,39)Polycyclic presentation:
2^9|a^2=gh,b^2=gi,c^2=gi,d^2=gi,e^2=ghi,f^2=gi,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, 8416158);