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 | |
4932 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 6 generators on 40 points:
(9,10,14,15)(11,13,17,21)(12,16,19,22)(18,20,23,24)(25,26)(27,34)(28,29)(30,32)(31,38)(33,35)(36,37)(39,40);(1,2,5,6)(3,7,8,4)(12,19)(16,22)(18,23)(20,24)(25,27,30,38)(26,31,32,34)(28,39,35,36)(29,37,33,40);(1,3)(2,4)(5,8)(6,7)(10,15)(11,17)(16,22)(18,23)(26,32)(28,35)(31,34)(36,39);(2,6)(4,7)(9,11)(10,13)(12,18)(14,17)(15,21)(16,20)(19,23)(22,24)(25,28,30,35)(26,33,32,29)(27,36,38,39)(31,37,34,40);(1,4)(2,3)(5,7)(6,8)(9,12)(10,16)(11,18)(13,20)(14,19)(15,22)(17,23)(21,24)(25,27)(26,34)(28,39)(29,40)(30,38)(31,32)(33,37)(35,36);(1,3)(2,7)(4,6)(5,8)(9,13)(10,11)(12,20)(14,21)(15,17)(16,18)(19,24)(22,23)(25,29,30,33)(26,35,32,28)(27,37,38,40)(31,36,34,39)Polycyclic presentation:
2^9|a^2=gh,b^2=gi,d^2=g,f^2=g,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,d^b=dgi,e^b=eh,d^c=dh,e^d=egi,f^e=fgi80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 10132690);