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 | |
4502 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)(9,10)(11,16)(12,15)(13,17)(14,18)(19,23)(20,21)(22,24)(25,26,30,29)(27,34,37,31)(28,33,38,32)(35,39,36,40);(1,3,5,8)(2,6,4,7)(9,11)(10,14)(12,19)(13,18)(15,22)(16,17)(20,23)(21,24)(25,27,30,37)(26,31,29,34)(28,36,38,35)(32,40,33,39);(11,18)(14,16)(19,24)(22,23)(25,28,30,38)(26,32,29,33)(27,35,37,36)(31,39,34,40);(1,2,5,4)(3,6,8,7)(9,12)(10,15)(11,19)(13,21)(14,22)(16,23)(17,20)(18,24)(25,28)(26,33)(27,36)(29,32)(30,38)(31,39)(34,40)(35,37);(1,4,5,2)(3,7,8,6)(9,10)(11,16)(12,20)(13,17)(14,18)(15,21)(19,22)(23,24)(25,27)(26,31)(28,35)(29,34)(30,37)(32,39)(33,40)(36,38);(9,11)(10,16)(12,19)(13,18)(14,17)(15,23)(20,22)(21,24)(25,29,30,26)(27,34,37,31)(28,32,38,33)(35,39,36,40)Polycyclic presentation:
2^9|a^2=ghi,b^2=ghi,c^2=h,d^2=gi,e^2=gi,f^2=h,b^a=bg,c^a=ch,e^a=eh,c^b=ci,e^b=ei,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, 7705264);