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 | |
4351 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 40 points:
(1,2,5,6)(3,7,8,4)(12,20)(16,21)(19,24)(22,23)(25,26,30,34)(27,31,37,40)(28,29,33,32)(35,36,39,38);(2,6)(3,8)(9,10)(11,15)(12,21)(13,17)(14,18)(16,20)(19,22)(23,24)(25,27,30,37)(26,31,34,40)(28,35,33,39)(29,36,32,38);(9,11)(10,13)(12,19)(14,17)(15,18)(16,22)(20,24)(21,23)(25,28,30,33)(26,32,34,29)(27,35,37,39)(31,38,40,36);(2,6)(4,7)(9,11)(10,15)(12,19)(13,18)(14,17)(16,23)(20,24)(21,22)(25,27)(26,31)(28,35)(29,36)(30,37)(32,38)(33,39)(34,40);(1,3)(2,4)(5,8)(6,7)(9,12,14,20)(10,16,18,21)(11,19,17,24)(13,22,15,23)(25,26)(27,31)(28,32)(29,33)(30,34)(35,38)(36,39)(37,40);(1,4)(2,3)(5,7)(6,8)(9,13,14,15)(10,17,18,11)(12,22,20,23)(16,24,21,19)(25,29,30,32)(26,33,34,28)(27,36,37,38)(31,39,40,35)Polycyclic presentation:
2^9|a^2=gh,b^2=h,c^2=h,e^2=i,f^2=hi,b^a=bg,c^a=ch,d^a=dg,e^a=eghi,c^b=ci,d^b=dh,e^b=egi,f^d=fgi83 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8413452);