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 | |
4467 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 40 points:
(2,5)(6,8)(9,10,14,15)(11,13,18,16)(12,20,23,17)(19,24,21,22)(25,26,31,30)(27,32,34,28)(29,37,39,33)(35,36,40,38);(3,7)(6,8)(10,15)(11,18)(17,20)(19,21)(25,27)(26,32)(28,30)(29,35)(31,34)(33,38)(36,37)(39,40);(1,2)(3,6)(4,5)(7,8)(10,15)(13,16)(17,20)(22,24)(26,30)(27,34)(33,37)(35,40);(1,3)(2,6)(4,7)(5,8)(9,11,14,18)(10,16,15,13)(12,21,23,19)(17,22,20,24)(25,28)(26,27)(29,38)(30,34)(31,32)(33,40)(35,37)(36,39);(1,2)(3,6)(4,5)(7,8)(9,12,14,23)(10,17,15,20)(11,19,18,21)(13,22,16,24)(25,29)(26,33)(27,35)(28,36)(30,37)(31,39)(32,38)(34,40);(9,13,14,16)(10,18,15,11)(12,22,23,24)(17,21,20,19)(25,30,31,26)(27,28,34,32)(29,37,39,33)(35,36,40,38)Polycyclic presentation:
2^9|a^2=gi,d^2=g,e^2=g,f^2=gi,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,f^c=fgi,e^d=egi86 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8230391);