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 | |
4389 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 40 points:
(9,10,14,15)(11,17,19,13)(12,16,20,22)(18,23,24,21)(25,26)(27,32)(28,34)(29,30)(31,33)(35,39)(36,40)(37,38);(1,2,5,7)(3,4,8,6)(10,15)(13,17)(16,22)(21,23)(25,27)(26,32)(28,35)(29,31)(30,33)(34,39)(36,37)(38,40);(1,3,5,8)(2,6,7,4)(12,20)(16,22)(18,24)(21,23)(26,33)(27,29)(28,36)(38,39);(9,11)(10,13)(12,18)(14,19)(15,17)(16,21)(20,24)(22,23)(25,28,31,36)(26,34,33,40)(27,35,29,37)(30,38,32,39);(1,3)(2,4)(5,8)(6,7)(9,12)(10,16)(11,18)(13,21)(14,20)(15,22)(17,23)(19,24)(25,29,31,27)(26,30,33,32)(28,37,36,35)(34,38,40,39);(1,4)(2,3)(5,6)(7,8)(9,13)(10,11)(12,21)(14,17)(15,19)(16,18)(20,23)(22,24)(25,30)(26,27)(28,38)(29,33)(31,32)(34,35)(36,39)(37,40)Polycyclic presentation:
2^9|a^2=g,b^2=hi,c^2=hi,d^2=h,e^2=h,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, 9157499);