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 | |
4946 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 6 generators on 40 points:
(1,2,5,6)(3,4,8,7)(9,10)(11,15)(12,16)(13,17)(14,18)(19,22)(20,23)(21,24)(25,26)(27,34)(28,33)(29,32)(30,35)(31,37)(36,39)(38,40);(2,6)(4,7)(11,18)(12,20)(14,15)(16,23)(25,27)(26,31)(28,36)(29,38)(30,37)(32,39)(33,40)(34,35);(9,11)(10,14)(12,19)(13,18)(15,17)(16,21)(20,24)(22,23)(25,28,30,29)(26,32,35,33)(27,36,37,38)(31,39,34,40);(1,3,5,8)(2,7,6,4)(9,11)(10,15)(12,19)(13,18)(14,17)(16,22)(20,24)(21,23)(25,29,30,28)(26,33,35,32)(27,36,37,38)(31,39,34,40);(1,4)(2,3)(5,7)(6,8)(9,12,13,20)(10,16,17,23)(11,19,18,24)(14,21,15,22)(25,26)(27,34)(28,32)(29,33)(30,35)(31,37)(36,40)(38,39);(1,3)(2,4)(5,8)(6,7)(9,10)(11,14)(12,16)(13,17)(15,18)(19,21)(20,23)(22,24)(25,27)(26,34)(28,36)(29,38)(30,37)(31,35)(32,40)(33,39)Polycyclic presentation:
2^9|a^2=ghi,c^2=hi,d^2=g,e^2=i,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, 8399999);