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 | |
4607 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 40 points:
(2,6)(7,8)(9,10)(11,16)(12,20)(13,15)(14,18)(17,22)(19,23)(21,24)(25,26,31,35)(27,32,30,34)(28,29,33,40)(36,37,38,39);(1,2,5,6)(3,7,4,8)(9,11)(10,14)(12,19)(13,18)(15,16)(17,23)(20,24)(21,22)(25,27)(26,32)(28,36)(29,37)(30,31)(33,38)(34,35)(39,40);(10,15)(11,18)(17,20)(19,21)(27,30)(32,34)(36,38)(37,39);(9,11)(10,16)(12,21)(13,18)(14,15)(17,23)(19,22)(20,24)(25,28)(26,29)(27,36)(30,38)(31,33)(32,37)(34,39)(35,40);(1,3,5,4)(2,7,6,8)(9,12,13,22)(10,17,15,20)(11,19,18,21)(14,23,16,24)(25,29)(26,33)(27,37)(28,35)(30,39)(31,40)(32,38)(34,36);(1,4,5,3)(2,8,6,7)(9,10)(11,16)(12,17)(13,15)(14,18)(19,24)(20,22)(21,23)(25,30,31,27)(26,34,35,32)(28,38,33,36)(29,39,40,37)Polycyclic presentation:
2^9|a^2=hi,b^2=gh,e^2=g,f^2=gi,b^a=bg,c^a=ch,e^a=eh,c^b=ci,f^b=fi,d^c=dh,f^c=fi,e^d=ei98 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7745845);