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 | |
4435 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)(3,7)(4,6)(5,8)(12,20)(16,23)(19,24)(21,22)(25,26,31,36)(27,30,35,32)(28,29,34,33)(37,39,40,38);(1,3)(2,5)(4,8)(6,7)(9,10)(11,15)(12,16)(13,17)(14,18)(19,22)(20,23)(21,24)(25,27)(26,32)(28,37)(29,38)(30,36)(31,35)(33,39)(34,40);(9,11)(10,13)(12,19)(14,17)(15,18)(16,21)(20,24)(22,23)(25,28,31,34)(26,33,36,29)(27,37,35,40)(30,38,32,39);(2,6)(5,7)(9,11)(10,15)(12,19)(13,18)(14,17)(16,22)(20,24)(21,23)(25,26)(27,30)(28,33)(29,34)(31,36)(32,35)(37,38)(39,40);(1,3)(2,5)(4,8)(6,7)(9,12,14,20)(10,16,18,23)(11,19,17,24)(13,21,15,22)(25,29,31,33)(26,34,36,28)(27,38,35,39)(30,37,32,40);(1,2)(3,5)(4,6)(7,8)(9,13,14,15)(10,17,18,11)(12,21,20,22)(16,24,23,19)(25,30)(26,35)(27,36)(28,38)(29,37)(31,32)(33,40)(34,39)Polycyclic presentation:
2^9|a^2=h,c^2=h,e^2=hi,f^2=i,b^a=bg,c^a=ch,d^a=dg,e^a=eghi,c^b=ci,d^b=dh,f^d=fgi89 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8366073);