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 | |
4509 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 40 points:
(1,2,5,3)(4,7,8,6)(13,21)(16,18)(19,24)(22,23)(25,26,30,32)(27,33,28,31)(29,37,38,34)(35,39,36,40);(2,3)(6,7)(9,10,14,11)(12,15,20,17)(13,16,21,18)(19,22,24,23)(25,27,30,28)(26,31,32,33)(29,35,38,36)(34,39,37,40);(1,3,5,2)(4,7,8,6)(10,11)(15,17)(16,18)(22,23)(26,32)(31,33)(34,37)(39,40);(1,4,5,8)(2,6,3,7)(9,11,14,10)(12,15,20,17)(13,18,21,16)(19,22,24,23)(25,26)(27,33)(28,31)(29,34)(30,32)(35,40)(36,39)(37,38);(9,12,14,20)(10,15,11,17)(13,19,21,24)(16,22,18,23)(25,28,30,27)(26,33,32,31)(29,36,38,35)(34,40,37,39);(9,13)(10,16)(11,18)(12,19)(14,21)(15,22)(17,23)(20,24)(25,29)(26,34)(27,35)(28,36)(30,38)(31,39)(32,37)(33,40)Polycyclic presentation:
2^9|a^2=g,b^2=gi,c^2=gh,d^2=i,e^2=gi,b^a=bg,c^a=ch,d^a=dg,e^a=eh,f^a=fgi,c^b=ci,d^b=dh,d^c=dgi,e^d=egi86 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8074600);