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 | |
4362 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 40 points:
(9,10,14,12)(11,17,19,15)(13,16,21,20)(18,23,24,22)(25,26)(27,29)(28,33)(30,38)(31,32)(34,40)(35,39)(36,37);(1,2,5,7)(3,4,8,6)(10,12)(15,17)(16,20)(22,23)(25,27,31,28)(26,29,32,33)(30,35,40,36)(34,37,38,39);(1,3)(2,6)(4,7)(5,8)(26,32)(27,28)(34,38)(35,36);(9,11)(10,15)(12,17)(13,18)(14,19)(16,22)(20,23)(21,24)(25,28,31,27)(26,33,32,29)(30,36,40,35)(34,39,38,37);(1,4,5,6)(2,3,7,8)(9,12,14,10)(11,17,19,15)(13,16,21,20)(18,22,24,23)(25,29,31,33)(26,27,32,28)(30,39,40,37)(34,36,38,35);(1,3)(2,4)(5,8)(6,7)(9,13,14,21)(10,16,12,20)(11,18,19,24)(15,22,17,23)(25,30,31,40)(26,34,32,38)(27,35,28,36)(29,37,33,39)Polycyclic presentation:
2^9|a^2=g,b^2=i,d^2=h,e^2=gi,f^2=gh,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eghi,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, 9750974);