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 | #34547 | C2 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
4411 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 40 points:
(4,8)(6,7)(9,10)(11,13)(12,20)(14,15)(16,18)(17,22)(19,23)(21,24)(25,26,31,36)(27,32,38,39)(28,35,30,33)(29,34,40,37);(1,2,5,3)(4,6,8,7)(10,15)(11,18)(17,20)(19,24)(25,27,31,38)(26,32,36,39)(28,34,30,37)(29,35,40,33);(2,3)(6,7)(9,11)(10,16)(12,19)(13,15)(14,18)(17,23)(20,21)(22,24)(25,28,31,30)(26,33,36,35)(27,37,38,34)(29,32,40,39);(1,3,5,2)(4,7,8,6)(9,11)(10,16)(12,19)(13,15)(14,18)(17,23)(20,21)(22,24)(25,29,31,40)(26,34,36,37)(27,33,38,35)(28,39,30,32);(9,12,14,22)(10,17,15,20)(11,19,18,24)(13,21,16,23)(25,30,31,28)(26,35,36,33)(27,37,38,34)(29,32,40,39);(1,4)(2,6)(3,7)(5,8)(9,13,14,16)(10,11,15,18)(12,21,22,23)(17,19,20,24)(25,26)(27,32)(28,33)(29,34)(30,35)(31,36)(37,40)(38,39)Polycyclic presentation:
2^9|a^2=gh,b^2=gi,c^2=gh,d^2=gi,e^2=h,f^2=g,b^a=bg,c^a=ch,d^a=dg,e^a=eh,f^a=fgi,c^b=ci,d^b=dh,d^c=dgi92 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8014686);