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 | |
4263 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 40 points:
(1,2,5,3)(4,7,8,6)(9,10,15,20)(11,18,13,16)(12,17,19,14)(21,22,23,24)(25,26,30,33)(27,31,28,35)(29,32,38,37)(34,39,36,40);(2,3)(6,7)(11,13)(16,18)(21,23)(22,24)(25,26)(27,31)(28,35)(29,32)(30,33)(34,39)(36,40)(37,38);(1,3,5,2)(4,7,8,6)(9,11,15,13)(10,16,20,18)(12,21,19,23)(14,22,17,24)(25,27,30,28)(26,31,33,35)(29,34,38,36)(32,39,37,40);(1,4,5,8)(2,6,3,7)(9,12)(10,17)(11,21)(13,23)(14,20)(15,19)(16,24)(18,22)(25,26)(27,31)(28,35)(29,37)(30,33)(32,38)(34,40)(36,39);(9,13,15,11)(10,18,20,16)(12,23,19,21)(14,24,17,22)(25,28,30,27)(26,31,33,35)(29,36,38,34)(32,39,37,40);(9,14,15,17)(10,19,20,12)(11,22,13,24)(16,23,18,21)(25,29)(26,32)(27,34)(28,36)(30,38)(31,39)(33,37)(35,40)Polycyclic presentation:
2^9|a^2=gh,c^2=gh,d^2=hi,e^2=gi,f^2=h,b^a=bg,c^a=ch,d^a=dg,e^a=egi,f^a=fh,c^b=ci,e^b=egi,e^d=eghi,f^d=fgi83 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 10349758);