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 | |
4495 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 40 points:
(2,4)(6,7)(9,10)(11,17)(12,16)(13,22)(14,19)(15,20)(18,23)(21,24)(25,26,30,34)(27,33,37,31)(28,38,29,32)(35,39,36,40);(1,2,5,4)(3,7,8,6)(9,11,15,12)(10,16,20,17)(13,21,23,19)(14,18,24,22)(25,27,30,37)(26,31,34,33)(28,35,29,36)(32,39,38,40);(1,3,5,8)(2,6,4,7)(9,11,15,12)(10,17,20,16)(13,19,23,21)(14,18,24,22)(25,28,30,29)(26,32,34,38)(27,35,37,36)(31,39,33,40);(1,4,5,2)(3,7,8,6)(9,12,15,11)(10,17,20,16)(13,21,23,19)(14,18,24,22)(25,26)(27,33)(28,32)(29,38)(30,34)(31,37)(35,40)(36,39);(9,13,15,23)(10,18,20,22)(11,19,12,21)(14,17,24,16)(25,27)(26,33)(28,35)(29,36)(30,37)(31,34)(32,40)(38,39);(9,14,15,24)(10,19,20,21)(11,18,12,22)(13,17,23,16)(25,29,30,28)(26,32,34,38)(27,36,37,35)(31,39,33,40)Polycyclic presentation:
2^9|a^2=h,b^2=g,c^2=g,d^2=gh,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, 8413253);