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 | |
4522 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 40 points:
(1,2,5,3)(4,7,8,6)(9,10,15,16)(11,17,12,21)(13,22,23,18)(14,19,24,20)(25,26)(27,34)(28,29)(30,32)(31,33)(35,39)(36,40)(37,38);(2,3)(6,7)(10,16)(13,23)(17,21)(19,20)(25,26)(27,34)(28,37)(29,38)(30,32)(31,33)(35,36)(39,40);(1,3,5,2)(4,7,8,6)(9,11,15,12)(10,17,16,21)(13,20,23,19)(14,18,24,22)(25,27,31,30)(26,32,33,34)(28,35,38,40)(29,36,37,39);(1,4,5,8)(2,6,3,7)(9,12,15,11)(10,17,16,21)(13,19,23,20)(14,18,24,22)(25,28)(26,29)(27,35)(30,40)(31,38)(32,36)(33,37)(34,39);(9,13)(10,18)(11,20)(12,19)(14,21)(15,23)(16,22)(17,24)(25,29)(26,28)(27,36)(30,39)(31,37)(32,35)(33,38)(34,40);(9,14,15,24)(10,19,16,20)(11,18,12,22)(13,21,23,17)(25,30,31,27)(26,32,33,34)(28,35,38,40)(29,39,37,36)Polycyclic presentation:
2^9|a^2=g,c^2=gh,d^2=g,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, 8403680);