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 | #42595 | C2 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
4369 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)(11,12)(13,17)(14,15)(16,19)(18,20)(21,22)(23,24)(25,26,32,36)(27,34,28,33)(29,35,30,31)(37,40,38,39);(1,2)(3,4)(5,6)(7,8)(10,15)(12,16)(17,22)(20,23)(25,27,32,28)(26,33,36,34)(29,38,30,37)(31,39,35,40);(2,6)(4,7)(13,21)(17,22)(18,24)(20,23)(25,28,32,27)(26,34,36,33)(29,38,30,37)(31,39,35,40);(9,11)(10,16)(12,15)(13,18)(14,19)(17,23)(20,22)(21,24)(25,29,32,30)(26,31,36,35)(27,37,28,38)(33,40,34,39);(1,3)(2,7)(4,6)(5,8)(9,12,14,16)(10,11,15,19)(13,20,21,23)(17,18,22,24)(25,30,32,29)(26,35,36,31)(27,38,28,37)(33,39,34,40);(1,4,5,7)(2,3,6,8)(9,13,14,21)(10,17,15,22)(11,18,19,24)(12,20,16,23)(25,31,32,35)(26,29,36,30)(27,39,28,40)(33,38,34,37)Polycyclic presentation:
2^9|a^2=h,b^2=h,c^2=h,d^2=h,e^2=g,f^2=gi,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,e^b=egi,f^c=fgi86 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8138657);