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 | |
4553 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 40 points:
(1,2,4,6)(3,5,7,8)(9,10,14,15)(11,19,22,16)(12,13,18,17)(20,21,24,23)(25,26,30,29)(27,32,34,31)(28,33,36,37)(35,39,40,38);(3,7)(5,8)(10,15)(12,18)(16,19)(20,24)(27,34)(28,36)(31,32)(33,37);(1,2)(3,5)(4,6)(7,8)(25,27)(26,31)(28,35)(29,32)(30,34)(33,38)(36,40)(37,39);(1,3)(2,5)(4,7)(6,8)(9,11,14,22)(10,16,15,19)(12,20,18,24)(13,23,17,21)(25,27)(26,32)(28,35)(29,31)(30,34)(33,39)(36,40)(37,38);(1,2)(3,5)(4,6)(7,8)(9,12,14,18)(10,17,15,13)(11,20,22,24)(16,21,19,23)(25,28,30,36)(26,33,29,37)(27,35,34,40)(31,38,32,39);(9,13,14,17)(10,18,15,12)(11,21,22,23)(16,20,19,24)(25,29,30,26)(27,32,34,31)(28,37,36,33)(35,39,40,38)Polycyclic presentation:
2^9|a^2=gh,d^2=g,e^2=gi,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, 8412882);