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 | |
4596 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)(11,17)(15,22)(18,20)(23,24)(25,26)(27,34)(28,32)(29,33)(30,31)(35,37)(36,40)(38,39);(2,3)(6,7)(9,10)(11,15)(12,19)(13,14)(16,21)(17,22)(18,24)(20,23)(25,27)(26,28)(29,35)(30,34)(31,32)(33,36)(37,38)(39,40);(1,3,5,2)(4,7,8,6)(10,14)(11,17)(16,19)(18,20)(26,30)(28,34)(33,38)(36,37);(1,4,5,8)(2,6,3,7)(9,11)(10,15)(12,20)(13,17)(14,22)(16,24)(18,21)(19,23)(25,28,31,34)(26,32,30,27)(29,37,39,36)(33,35,38,40);(9,12,13,21)(10,16,14,19)(11,18,17,20)(15,23,22,24)(25,29,31,39)(26,33,30,38)(27,35,32,40)(28,36,34,37);(9,10)(11,15)(12,16)(13,14)(17,22)(18,23)(19,21)(20,24)(25,30,31,26)(27,34,32,28)(29,38,39,33)(35,37,40,36)Polycyclic presentation:
2^9|a^2=gh,c^2=gh,d^2=g,e^2=gi,f^2=h,b^a=bg,c^a=ch,d^a=di,f^a=fh,c^b=ci,e^b=eghi,d^c=dgi,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, 8838616);