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 | |
4376 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)(3,4)(5,6)(7,8)(11,17)(15,22)(18,20)(23,24)(25,26,31,32)(27,35,36,33)(28,38,37,34)(29,30,39,40);(2,6)(3,7)(9,10)(11,15)(12,19)(13,14)(16,21)(17,22)(18,24)(20,23)(26,32)(27,36)(29,39)(34,38);(1,3,5,7)(2,4,6,8)(10,14)(11,17)(16,19)(18,20)(25,27,31,36)(26,33,32,35)(28,29,37,39)(30,38,40,34);(1,4)(2,7)(3,6)(5,8)(9,11)(10,15)(12,20)(13,17)(14,22)(16,24)(18,21)(19,23)(25,28,31,37)(26,34,32,38)(27,29,36,39)(30,35,40,33);(9,12,13,21)(10,16,14,19)(11,18,17,20)(15,23,22,24)(25,29)(26,30)(27,37)(28,36)(31,39)(32,40)(33,38)(34,35);(9,10)(11,15)(12,16)(13,14)(17,22)(18,23)(19,21)(20,24)(25,30)(26,29)(27,38)(28,35)(31,40)(32,39)(33,37)(34,36)Polycyclic presentation:
2^9|a^2=h,c^2=g,d^2=h,e^2=gi,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, 8838204);