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 | |
4467 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 40 points:
(9,10,14,16)(11,18,22,15)(12,21,23,17)(13,20,24,19)(25,26)(27,30)(28,35)(29,32)(31,37)(33,40)(34,39)(36,38);(1,2,5,7)(3,4,8,6)(9,11)(10,15)(12,20)(13,21)(14,22)(16,18)(17,24)(19,23)(25,26)(27,30)(28,31)(29,32)(33,38)(34,39)(35,37)(36,40);(1,3)(2,6)(4,7)(5,8)(10,16)(12,23)(15,18)(19,20)(25,26,29,32)(27,30,34,39)(28,31,37,35)(33,38,36,40);(9,12)(10,17)(11,20)(13,18)(14,23)(15,24)(16,21)(19,22)(25,26)(27,30)(28,31)(29,32)(33,38)(34,39)(35,37)(36,40);(1,4,5,6)(2,3,7,8)(9,11)(10,18)(12,20)(13,17)(14,22)(15,16)(19,23)(21,24)(25,27,29,34)(26,30,32,39)(28,36,37,33)(31,40,35,38);(1,3)(2,4)(5,8)(6,7)(9,13)(10,19)(11,21)(12,18)(14,24)(15,23)(16,20)(17,22)(25,28,29,37)(26,31,32,35)(27,33,34,36)(30,38,39,40)Polycyclic presentation:
2^9|a^2=g,b^2=ghi,c^2=gh,e^2=i,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, 9750873);