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 | |
4345 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 40 points:
(1,2,5,3)(4,7,8,6)(9,10,15,16)(11,17,12,22)(13,14,19,18)(20,21,23,24)(25,26)(27,30)(28,33)(29,34)(31,35)(32,37)(36,38)(39,40);(2,3)(6,7)(10,16)(13,19)(17,22)(20,23)(26,27)(28,35)(32,34)(36,40);(1,3,5,2)(4,7,8,6)(9,11,15,12)(10,17,16,22)(13,20,19,23)(14,21,18,24)(25,27,30,26)(28,33,35,31)(29,34,37,32)(36,39,40,38);(1,4,5,8)(2,6,3,7)(9,12,15,11)(10,17,16,22)(13,23,19,20)(14,21,18,24)(25,28)(26,31)(27,33)(29,36)(30,35)(32,38)(34,39)(37,40);(9,13,15,19)(10,18,16,14)(11,20,12,23)(17,24,22,21)(25,29,30,37)(26,32,27,34)(28,36,35,40)(31,38,33,39);(9,14,15,18)(10,19,16,13)(11,21,12,24)(17,23,22,20)Polycyclic presentation:
2^9|a^2=gh,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, 8413252);