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 | |
4245 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)(11,18)(17,20)(19,24)(25,26)(27,30)(28,29)(31,32)(33,39)(34,35)(36,38)(37,40);(2,3)(6,7)(9,10,14,15)(11,16,18,13)(12,20,22,17)(19,23,24,21)(25,27,31,33)(26,30,32,39)(28,34,36,37)(29,35,38,40);(1,2,5,3)(4,6,8,7)(26,32)(27,33)(28,36)(35,40);(9,11)(10,16)(12,19)(13,15)(14,18)(17,21)(20,23)(22,24)(25,28)(26,29)(27,34)(30,35)(31,36)(32,38)(33,37)(39,40);(1,3,5,2)(4,6,8,7)(9,12,14,22)(10,17,15,20)(11,19,18,24)(13,23,16,21)(25,29)(26,28)(27,35)(30,34)(31,38)(32,36)(33,40)(37,39);(1,4)(2,6)(3,7)(5,8)(9,13,14,16)(10,11,15,18)(12,21,22,23)(17,24,20,19)(25,30)(26,33)(27,32)(28,35)(29,37)(31,39)(34,38)(36,40)Polycyclic presentation:
2^9|b^2=gh,c^2=hi,e^2=ghi,f^2=g,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, 9755876);