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 | |
4392 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 40 points:
(1,2,5,4)(3,7,8,6)(25,26)(27,29)(28,33)(30,32)(31,34)(35,38)(36,39)(37,40);(1,3,5,8)(2,6,4,7)(9,10,14,15)(11,13,17,22)(12,16,21,23)(18,20,19,24)(25,27,30,34)(26,31,32,29)(28,35,36,40)(33,37,39,38);(1,2,5,4)(3,6,8,7)(10,15)(13,22)(16,23)(20,24)(26,32)(27,34)(28,36)(37,38);(9,11,14,17)(10,13,15,22)(12,19,21,18)(16,24,23,20)(25,28)(26,33)(27,35)(29,38)(30,36)(31,37)(32,39)(34,40);(1,2,5,4)(3,6,8,7)(9,12)(10,16)(11,18)(13,20)(14,21)(15,23)(17,19)(22,24)(25,29,30,31)(26,34,32,27)(28,37,36,38)(33,35,39,40);(1,4,5,2)(3,7,8,6)(9,13)(10,17)(11,15)(12,20)(14,22)(16,19)(18,23)(21,24)(25,26)(27,31)(28,33)(29,34)(30,32)(35,37)(36,39)(38,40)Polycyclic presentation:
2^9|a^2=gh,b^2=ghi,c^2=gh,d^2=hi,e^2=g,f^2=gh,b^a=bg,c^a=ch,e^a=eh,c^b=ci,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, 7679196);