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 | |
4704 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)(11,18)(12,15)(13,16)(14,17)(19,24)(20,21)(22,23)(25,26)(27,30)(28,37)(29,33)(31,32)(34,40)(35,36)(38,39);(2,3)(6,7)(9,11,14,16)(10,13,17,18)(12,19,20,23)(15,22,21,24)(25,27,31,34)(26,30,32,40)(28,35,33,39)(29,38,37,36);(1,3,5,2)(4,7,8,6)(12,20)(15,21)(19,23)(22,24)(26,32)(27,34)(28,33)(36,38);(1,4,5,8)(2,6,3,7)(9,12,14,20)(10,15,17,21)(11,19,16,23)(13,22,18,24)(25,28)(26,29)(27,35)(30,38)(31,33)(32,37)(34,39)(36,40);(9,10)(11,13)(12,21)(14,17)(15,20)(16,18)(19,24)(22,23)(25,29)(26,33)(27,36)(28,32)(30,35)(31,37)(34,38)(39,40);(9,13)(10,16)(11,17)(12,22)(14,18)(15,23)(19,21)(20,24)(25,30)(26,34)(27,32)(28,38)(29,39)(31,40)(33,36)(35,37)Polycyclic presentation:
2^9|a^2=hi,b^2=gi,c^2=hi,d^2=g,b^a=bg,c^a=ch,d^a=di,f^a=fh,c^b=ci,e^b=eh,d^c=dgi,e^d=egi,f^e=fgi83 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 9096749);