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 | #36676 | C2 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
4224 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 40 points:
(3,4)(6,7)(9,10,15,17)(11,20,14,16)(12,18,22,24)(13,21,23,19)(25,26)(27,30)(28,33)(29,34)(31,35)(32,37)(36,38)(39,40);(1,2,5,8)(3,6,4,7)(9,11,15,14)(10,16,17,20)(12,19,22,21)(13,18,23,24)(26,27)(28,35)(32,34)(36,40);(1,3,5,4)(2,6,8,7)(10,17)(12,22)(16,20)(19,21)(25,27,30,26)(28,33,35,31)(29,34,37,32)(36,39,40,38);(1,2)(3,6)(4,7)(5,8)(9,12,15,22)(10,18,17,24)(11,21,14,19)(13,20,23,16)(25,28)(26,31)(27,33)(29,36)(30,35)(32,38)(34,39)(37,40);(1,4,5,3)(2,7,8,6)(9,13)(10,19)(11,18)(12,20)(14,24)(15,23)(16,22)(17,21)(25,29)(26,32)(27,34)(28,36)(30,37)(31,38)(33,39)(35,40);(9,14,15,11)(10,16,17,20)(12,21,22,19)(13,18,23,24)Polycyclic presentation:
2^9|a^2=gi,b^2=hi,c^2=ghi,d^2=gi,e^2=gh,f^2=gi,b^a=bg,c^a=ch,d^a=di,e^a=eh,c^b=ci,d^b=dh,d^c=dgi,f^d=fgi,f^e=fgi83 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8658757);