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 | |
4420 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 40 points:
(1,2,5,4)(3,7,8,6)(25,26)(27,28)(29,37)(30,31)(32,35)(33,38)(34,39)(36,40);(1,3,5,8)(2,6,4,7)(9,10,14,15)(11,16,21,24)(12,17,22,20)(13,18,23,19)(26,31)(28,32)(33,37)(36,39);(10,15)(13,23)(16,24)(17,20)(25,27,30,35)(26,32,31,28)(29,34,38,40)(33,39,37,36);(1,2,5,4)(3,6,8,7)(9,11)(10,16)(12,18)(13,20)(14,21)(15,24)(17,23)(19,22)(25,27)(26,28)(29,34)(30,35)(31,32)(33,36)(37,39)(38,40);(1,4,5,2)(3,7,8,6)(9,12)(10,17)(11,19)(13,16)(14,22)(15,20)(18,21)(23,24)(25,28)(26,27)(29,36)(30,32)(31,35)(33,34)(37,40)(38,39);(9,13)(10,18)(11,20)(12,16)(14,23)(15,19)(17,21)(22,24)(25,29,30,38)(26,33,31,37)(27,34,35,40)(28,36,32,39)Polycyclic presentation:
2^9|a^2=gh,b^2=ghi,c^2=h,d^2=gh,e^2=gh,f^2=h,b^a=bg,c^a=ch,f^a=fh,c^b=ci,e^b=eh,d^c=dh,f^c=fi,e^d=ei92 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7908188);