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 | |
4395 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)(3,6)(4,7)(5,8)(9,10,15,14)(11,18,13,16)(12,20,22,17)(19,23,21,24)(28,36)(32,39)(35,40)(37,38);(1,3,5,7)(2,6,8,4)(9,11,15,13)(10,16,14,18)(12,19,22,21)(17,23,20,24)(25,26)(27,31)(28,32)(29,33)(30,34)(35,38)(36,39)(37,40);(1,3)(2,4)(5,7)(6,8)(25,27)(26,29)(28,35)(30,33)(31,34)(32,37)(36,40)(38,39);(1,3)(2,6)(4,8)(5,7)(9,12,15,22)(10,17,14,20)(11,19,13,21)(16,23,18,24)(25,27)(26,31)(28,35)(29,34)(30,33)(32,38)(36,40)(37,39);(9,13,15,11)(10,18,14,16)(12,21,22,19)(17,24,20,23)(25,28,30,36)(26,32,34,39)(27,35,33,40)(29,37,31,38);(1,4,5,6)(2,7,8,3)(9,14,15,10)(11,18,13,16)(12,17,22,20)(19,23,21,24)(25,29,30,31)(26,33,34,27)(28,37,36,38)(32,40,39,35)Polycyclic presentation:
2^9|a^2=g,b^2=gh,d^2=g,e^2=ghi,f^2=gi,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, 8365996);