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 | |
4642 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 40 points:
(2,5)(6,8)(9,10,14,12)(11,15,18,19)(13,16,21,20)(17,22,23,24)(25,26)(27,31)(28,38)(29,33)(30,34)(32,36)(35,37)(39,40);(3,7)(6,8)(10,12)(15,19)(16,20)(22,24)(27,35)(29,39)(31,37)(33,40);(1,2)(3,6)(4,5)(7,8)(13,21)(16,20)(17,23)(22,24)(25,27,30,35)(26,31,34,37)(28,29,36,39)(32,40,38,33);(9,11,14,18)(10,15,12,19)(13,17,21,23)(16,22,20,24)(25,28)(26,32)(27,29)(30,36)(31,40)(33,37)(34,38)(35,39);(1,3)(2,6)(4,7)(5,8)(9,12,14,10)(11,15,18,19)(13,20,21,16)(17,22,23,24)(25,29)(26,33)(27,36)(28,35)(30,39)(31,32)(34,40)(37,38);(1,2)(3,6)(4,5)(7,8)(9,13)(10,16)(11,17)(12,20)(14,21)(15,22)(18,23)(19,24)(25,26)(27,37)(28,32)(29,33)(30,34)(31,35)(36,38)(39,40)Polycyclic presentation:
2^9|a^2=g,c^2=i,d^2=g,e^2=g,b^a=bg,c^a=ch,d^a=di,f^a=fh,c^b=ci,e^b=eghi,f^c=fgi,e^d=egi80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8833868);