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 | |
4542 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 40 points:
(1,2,5,6)(3,8,4,7)(9,10,14,17)(11,18,13,15)(12,20,22,16)(19,23,21,24)(25,26,30,34)(27,32,35,31)(28,29,33,39)(36,38,40,37);(2,6)(7,8)(9,11,14,13)(10,15,17,18)(12,19,22,21)(16,23,20,24)(27,35)(31,32)(36,40)(37,38);(1,3,5,4)(2,7,6,8)(25,27)(26,31)(28,36)(29,37)(30,35)(32,34)(33,40)(38,39);(1,2)(3,7)(4,8)(5,6)(9,10)(11,15)(12,16)(13,18)(14,17)(19,23)(20,22)(21,24)(25,27)(26,32)(28,36)(29,38)(30,35)(31,34)(33,40)(37,39);(9,12)(10,16)(11,19)(13,21)(14,22)(15,23)(17,20)(18,24)(25,28,30,33)(26,29,34,39)(27,36,35,40)(31,37,32,38);(1,4,5,3)(2,7,6,8)(9,13,14,11)(10,15,17,18)(12,21,22,19)(16,23,20,24)(25,29)(26,33)(27,37)(28,34)(30,39)(31,40)(32,36)(35,38)Polycyclic presentation:
2^9|a^2=gi,b^2=ghi,c^2=hi,e^2=i,f^2=g,b^a=bg,c^a=ch,d^a=dg,e^a=eghi,c^b=ci,d^b=dh,f^d=fgi89 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8363176);