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 | |
4474 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 40 points:
(9,10)(11,18)(12,22)(13,20)(14,16)(15,21)(17,23)(19,24)(25,26,31,33)(27,32,34,29)(28,30,37,40)(35,36,39,38);(2,3)(6,7)(9,11)(10,15)(12,20)(13,17)(14,21)(16,18)(19,23)(22,24)(25,27,31,34)(26,32,33,29)(28,35,37,39)(30,36,40,38);(1,2,5,3)(4,6,8,7)(10,16)(12,23)(15,18)(19,20)(26,33)(27,34)(28,37)(36,38);(9,12)(10,17)(11,20)(13,15)(14,23)(16,22)(18,24)(19,21)(25,28)(26,30)(27,35)(29,38)(31,37)(32,36)(33,40)(34,39);(1,3,5,2)(4,6,8,7)(9,11)(10,18)(12,20)(13,22)(14,21)(15,16)(17,24)(19,23)(25,29)(26,27)(28,38)(30,35)(31,32)(33,34)(36,37)(39,40);(1,4)(2,6)(3,7)(5,8)(9,13,14,24)(10,19,16,20)(11,17,21,22)(12,15,23,18)(25,30)(26,28)(27,36)(29,39)(31,40)(32,35)(33,37)(34,38)Polycyclic presentation:
2^9|a^2=gh,b^2=gh,c^2=ghi,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, 9755925);