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 | |
4292 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)(9,10,15,17)(11,19,14,16)(12,18,22,13)(20,24,23,21)(25,26)(27,30)(28,29)(31,32)(33,37)(34,40)(35,38)(36,39);(1,3)(2,6)(4,7)(5,8)(9,11,15,14)(10,16,17,19)(12,20,22,23)(13,24,18,21)(25,27,31,34)(26,30,32,40)(28,35,37,39)(29,38,33,36);(10,17)(12,22)(16,19)(20,23)(26,32)(27,34)(28,37)(36,38);(9,12)(10,18)(11,20)(13,17)(14,23)(15,22)(16,21)(19,24)(25,28)(26,33)(27,35)(29,32)(30,36)(31,37)(34,39)(38,40);(1,3)(2,7)(4,6)(5,8)(9,13,15,18)(10,12,17,22)(11,21,14,24)(16,23,19,20)(25,29,31,33)(26,28,32,37)(27,36,34,38)(30,39,40,35);(1,4,5,2)(3,7,8,6)(9,14,15,11)(10,19,17,16)(12,23,22,20)(13,21,18,24)(25,30)(26,34)(27,32)(28,36)(29,35)(31,40)(33,39)(37,38)Polycyclic presentation:
2^9|a^2=g,b^2=h,e^2=h,f^2=g,b^a=bg,c^a=ch,d^a=di,f^a=fh,c^b=ci,e^b=eh,d^c=dh,f^e=fgi83 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 9623527);