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 | |
4508 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)(3,4)(5,6)(7,8)(12,20)(16,23)(19,24)(21,22)(25,26,29,32)(27,30,35,33)(28,31,37,40)(34,38,36,39);(2,6)(3,7)(9,10)(11,15)(12,16)(13,17)(14,18)(19,22)(20,23)(21,24)(25,26,29,32)(27,33,35,30)(28,31,37,40)(34,39,36,38);(1,3,5,7)(2,4,6,8)(9,11)(10,13)(12,19)(14,17)(15,18)(16,21)(20,24)(22,23)(25,26)(27,30)(28,31)(29,32)(33,35)(34,38)(36,39)(37,40);(1,4)(2,7)(3,6)(5,8)(9,11)(10,15)(12,19)(13,18)(14,17)(16,22)(20,24)(21,23)(25,27,29,35)(26,30,32,33)(28,36,37,34)(31,39,40,38);(9,12,14,20)(10,16,18,23)(11,19,17,24)(13,21,15,22);(9,13,14,15)(10,17,18,11)(12,21,20,22)(16,24,23,19)(25,28,29,37)(26,31,32,40)(27,34,35,36)(30,38,33,39)Polycyclic presentation:
2^9|a^2=h,b^2=h,c^2=g,d^2=h,e^2=ghi,f^2=gi,b^a=bg,c^a=ch,d^a=dg,e^a=eghi,c^b=ci,d^b=dh,f^d=fgi89 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8366313);