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 | |
4524 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 40 points:
(10,15)(11,17)(16,22)(19,23)(25,26,30,31)(27,28,35,32)(29,37,39,33)(34,40,38,36);(1,2)(3,7)(4,5)(6,8)(9,10)(11,18)(12,17)(13,16)(14,15)(19,24)(20,23)(21,22)(26,31)(28,32)(33,37)(36,40);(2,5)(3,8)(11,17)(12,18)(13,21)(16,22)(25,27,30,35)(26,32,31,28)(29,38,39,34)(33,36,37,40);(1,3,4,8)(2,6,5,7)(9,11,14,17)(10,12,15,18)(13,19,21,23)(16,20,22,24)(25,28,30,32)(26,27,31,35)(29,36,39,40)(33,34,37,38);(1,3)(2,6)(4,8)(5,7)(9,12)(10,11)(13,20)(14,18)(15,17)(16,19)(21,24)(22,23)(25,27)(26,28)(29,34)(30,35)(31,32)(33,36)(37,40)(38,39);(1,2)(3,6)(4,5)(7,8)(9,13,14,21)(10,16,15,22)(11,19,17,23)(12,20,18,24)(25,29,30,39)(26,33,31,37)(27,34,35,38)(28,36,32,40)Polycyclic presentation:
2^9|a^2=h,c^2=h,d^2=gi,f^2=g,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,d^b=dgi,e^b=eghi,d^c=dgi,e^c=egi,f^c=fgi,e^d=egi92 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8969421);