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 |
| prime radical | #36710 | C2 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
4458 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 40 points:
(2,4)(6,7)(9,10)(11,14)(12,21)(13,17)(15,18)(16,20)(19,24)(22,23)(25,26,31,36)(27,35,30,32)(28,29,34,33)(37,40,39,38);(1,2,5,4)(3,7,8,6)(9,11)(10,14)(12,13)(15,19)(16,22)(17,21)(18,24)(20,23)(25,27,31,30)(26,32,36,35)(28,39,34,37)(29,40,33,38);(1,3,5,8)(2,6,4,7)(11,19)(12,20)(14,24)(16,21)(25,28,31,34)(26,33,36,29)(27,37,30,39)(32,40,35,38);(1,4,5,2)(3,7,8,6)(9,12)(10,16)(11,13)(14,22)(15,20)(17,24)(18,21)(19,23)(25,26)(27,32)(28,33)(29,34)(30,35)(31,36)(37,40)(38,39);(9,13)(10,17)(11,20)(12,19)(14,16)(15,23)(18,22)(21,24)(25,29,31,33)(26,34,36,28)(27,38,30,40)(32,39,35,37);(9,14)(10,11)(12,22)(13,16)(15,24)(17,20)(18,19)(21,23)(25,30,31,27)(26,35,36,32)(28,39,34,37)(29,40,33,38)Polycyclic presentation:
2^9|a^2=h,b^2=g,c^2=g,d^2=gh,e^2=h,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, 8835803);