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 | |
4756 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)(9,10,14,15)(11,18,17,13)(12,16,20,22)(19,24,23,21)(25,26,31,28)(27,32,29,33)(30,34,38,36)(35,39,37,40);(2,3)(6,7)(10,15)(12,20)(13,18)(19,23)(25,27,31,29)(26,32,28,33)(30,37,38,35)(34,40,36,39);(1,3,5,2)(4,7,8,6)(10,15)(11,17)(12,20)(21,24)(27,29)(32,33)(35,37)(39,40);(1,4,5,8)(2,6,3,7)(9,11,14,17)(10,13,15,18)(12,19,20,23)(16,21,22,24)(25,28,31,26)(27,33,29,32)(30,36,38,34)(35,40,37,39);(9,12)(10,16)(11,19)(13,21)(14,20)(15,22)(17,23)(18,24)(25,29,31,27)(26,33,28,32)(30,37,38,35)(34,40,36,39);(9,13)(10,17)(11,15)(12,21)(14,18)(16,23)(19,22)(20,24)(25,30,31,38)(26,34,28,36)(27,35,29,37)(32,39,33,40)Polycyclic presentation:
2^9|a^2=hi,b^2=ghi,c^2=gh,d^2=hi,e^2=ghi,f^2=ghi,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,f^b=fgi,d^c=dh,e^c=egi80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 9275344);