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 | |
4393 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)(9,10,14,12)(11,17,18,15)(13,20,21,16)(19,22,24,23)(25,26,31,28)(27,32,34,29)(30,33,39,36)(35,40,38,37);(2,3)(6,7)(10,12)(13,21)(15,17)(19,24)(25,27,31,34)(26,32,28,29)(30,38,39,35)(33,37,36,40);(1,3,5,2)(4,7,8,6)(10,12)(11,18)(16,20)(19,24)(27,34)(29,32)(35,38)(37,40);(1,4,5,8)(2,6,3,7)(9,11,14,18)(10,15,12,17)(13,19,21,24)(16,22,20,23)(25,28,31,26)(27,29,34,32)(30,36,39,33)(35,37,38,40);(9,12,14,10)(11,17,18,15)(13,20,21,16)(19,23,24,22)(25,29)(26,27)(28,34)(30,37)(31,32)(33,35)(36,38)(39,40);(9,13,14,21)(10,16,12,20)(11,19,18,24)(15,22,17,23)(25,30,31,39)(26,33,28,36)(27,35,34,38)(29,37,32,40)Polycyclic presentation:
2^9|a^2=hi,b^2=ghi,c^2=gh,d^2=hi,e^2=h,f^2=gi,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, 9240512);