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 | |
4659 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 40 points:
(2,5)(3,7)(9,10)(11,18)(12,15)(13,21)(14,16)(17,22)(19,23)(20,24)(25,26)(27,32)(28,31)(29,33)(30,36)(34,38)(35,37)(39,40);(1,2,6,5)(3,8,7,4)(9,11,14,12)(10,15,16,18)(13,19,22,20)(17,23,21,24)(25,27)(26,32)(28,29)(30,35)(31,33)(34,39)(36,37)(38,40);(1,3)(2,4)(5,8)(6,7)(10,16)(13,22)(15,18)(19,20)(26,28)(27,33)(30,38)(37,39);(1,2,6,5)(3,4,7,8)(9,10)(11,15)(12,18)(13,17)(14,16)(19,23)(20,24)(21,22)(25,28,31,26)(27,29,33,32)(30,36,38,34)(35,37,40,39);(1,4,6,8)(2,7,5,3)(9,12,14,11)(10,15,16,18)(13,20,22,19)(17,23,21,24)(25,29)(26,33)(27,28)(30,37)(31,32)(34,40)(35,36)(38,39);(1,5,6,2)(3,8,7,4)(9,13,14,22)(10,17,16,21)(11,19,12,20)(15,23,18,24)(25,30,31,38)(26,34,28,36)(27,35,33,40)(29,37,32,39)Polycyclic presentation:
2^9|b^2=g,d^2=i,e^2=g,f^2=h,b^a=bg,c^a=ch,d^a=di,f^a=fh,c^b=ci,e^b=eghi,d^c=dgi,f^c=fgi,e^d=egi80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8837543);