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 | |
4717 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 40 points:
(1,2,5,4)(3,7,8,6)(9,10,15,20)(11,21,23,16)(12,18,13,17)(14,22,24,19)(27,34)(29,37)(31,32)(33,36);(1,3)(2,6)(4,7)(5,8)(9,11)(10,16)(12,22)(13,19)(14,18)(15,23)(17,24)(20,21)(25,26)(27,32)(28,30)(29,33)(31,34)(35,39)(36,37)(38,40);(9,12,15,13)(10,17,20,18)(11,22,23,19)(14,16,24,21)(25,27)(26,31)(28,32)(29,35)(30,34)(33,38)(36,39)(37,40);(2,4)(6,7)(9,13,15,12)(10,18,20,17)(11,22,23,19)(14,16,24,21)(25,28,30,26)(27,32,34,31)(29,36,37,33)(35,39,40,38);(1,4,5,2)(3,7,8,6)(9,14,15,24)(10,19,20,22)(11,18,23,17)(12,16,13,21)(25,27)(26,32)(28,31)(29,35)(30,34)(33,39)(36,38)(37,40);(1,3)(2,6)(4,7)(5,8)(9,10)(11,16)(12,17)(13,18)(14,19)(15,20)(21,23)(22,24)(25,29)(26,33)(27,35)(28,36)(30,37)(31,38)(32,39)(34,40)Polycyclic presentation:
2^9|a^2=g,c^2=hi,d^2=h,e^2=g,b^a=bg,c^a=ch,d^a=dg,e^a=eh,f^a=fgi,c^b=ci,d^b=dh,e^d=egi86 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8044802);