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 | |
4721 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)(10,14)(11,17)(12,19)(23,24)(25,26,30,34)(27,35,38,31)(28,39,37,32)(29,33,36,40);(2,3)(6,7)(9,10,13,14)(11,15,17,21)(12,16,19,22)(18,23,20,24)(25,27,30,38)(26,31,34,35)(28,29,37,36)(32,33,39,40);(1,3,5,2)(4,7,8,6)(9,11)(10,15)(12,18)(13,17)(14,21)(16,23)(19,20)(22,24)(25,28)(26,32)(27,36)(29,38)(30,37)(31,40)(33,35)(34,39);(1,4)(2,6)(3,7)(5,8)(9,11)(10,15)(12,20)(13,17)(14,21)(16,24)(18,19)(22,23)(25,28)(26,32)(27,29)(30,37)(31,33)(34,39)(35,40)(36,38);(9,10)(11,15)(12,16)(13,14)(17,21)(18,23)(19,22)(20,24)(25,29)(26,33)(27,37)(28,38)(30,36)(31,39)(32,35)(34,40);(9,12)(10,16)(11,18)(13,19)(14,22)(15,23)(17,20)(21,24)(25,26)(27,31)(28,32)(29,33)(30,34)(35,38)(36,40)(37,39)Polycyclic presentation:
2^9|a^2=i,b^2=h,c^2=gh,b^a=bg,c^a=ch,d^a=dg,e^a=egi,f^a=fh,c^b=ci,e^b=egi,e^d=eghi,f^d=fgi83 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 10319420);