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 | |
4318 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 40 points:
(1,2,5,3)(4,7,8,6)(26,29)(27,28)(33,37)(35,36);(2,3)(6,7)(9,10,14,15)(11,13,17,22)(12,16,21,23)(18,20,19,24)(25,26,31,29)(27,32,28,34)(30,33,38,37)(35,39,36,40);(1,3,5,2)(4,7,8,6)(10,15)(13,22)(16,23)(20,24)(25,27,31,28)(26,32,29,34)(30,36,38,35)(33,40,37,39);(1,4)(2,6)(3,7)(5,8)(9,11,14,17)(10,13,15,22)(12,19,21,18)(16,24,23,20)(25,28,31,27)(26,32,29,34)(30,36,38,35)(33,39,37,40);(9,12)(10,16)(11,18)(13,20)(14,21)(15,23)(17,19)(22,24)(25,29,31,26)(27,34,28,32)(30,37,38,33)(35,40,36,39);(9,13)(10,17)(11,15)(12,20)(14,22)(16,19)(18,23)(21,24)(25,30)(26,33)(27,35)(28,36)(29,37)(31,38)(32,39)(34,40)Polycyclic presentation:
2^9|a^2=gh,b^2=gi,c^2=g,d^2=gi,e^2=h,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, 8202740);