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 | |
4375 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 40 points:
(9,10,14,15)(11,12,16,20)(13,21,22,17)(18,24,23,19)(25,26)(27,34)(28,33)(29,32)(30,35)(31,36)(37,38)(39,40);(1,2)(3,8)(4,6)(5,7)(9,11)(10,12)(13,18)(14,16)(15,20)(17,19)(21,24)(22,23)(25,27,31,29)(26,32,36,34)(28,30,37,39)(33,40,38,35);(1,3)(2,4)(5,6)(7,8)(10,15)(11,16)(13,22)(19,24)(25,28,31,37)(26,33,36,38)(27,30,29,39)(32,40,34,35);(3,6)(4,8)(9,11)(10,12)(13,18)(14,16)(15,20)(17,19)(21,24)(22,23)(25,29,31,27)(26,34,36,32)(28,30,37,39)(33,40,38,35);(1,4,5,8)(2,6,7,3)(9,12)(10,16)(11,15)(13,19)(14,20)(17,23)(18,21)(22,24)(25,30)(26,35)(27,37)(28,29)(31,39)(32,33)(34,38)(36,40);(1,2)(3,8)(4,6)(5,7)(9,13,14,22)(10,17,15,21)(11,18,16,23)(12,19,20,24)(25,26)(27,32)(28,38)(29,34)(30,35)(31,36)(33,37)(39,40)Polycyclic presentation:
2^9|a^2=h,b^2=g,c^2=g,d^2=g,e^2=hi,f^2=h,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,d^c=dgi,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, 8821473);