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 | |
4440 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 40 points:
(1,2,5,3)(4,7,8,6)(9,10,14,15)(11,13,17,21)(12,16,19,22)(18,20,23,24)(25,26)(27,35)(28,31)(29,33)(30,34)(32,36)(37,39)(38,40);(2,3)(6,7)(12,19)(16,22)(18,23)(20,24)(25,27)(26,32)(28,35)(29,30)(31,36)(33,38)(34,37)(39,40);(1,3,5,2)(4,7,8,6)(10,15)(11,17)(16,22)(18,23)(29,39)(30,40)(33,37)(34,38);(1,4,5,8)(2,6,3,7)(9,11)(10,13)(12,18)(14,17)(15,21)(16,20)(19,23)(22,24)(25,28,31,26)(27,35,36,32)(29,33,39,37)(30,38,40,34);(9,12)(10,16)(11,18)(13,20)(14,19)(15,22)(17,23)(21,24)(25,29)(26,33)(27,30)(28,37)(31,39)(32,38)(34,35)(36,40);(9,13)(10,11)(12,20)(14,21)(15,17)(16,18)(19,24)(22,23)(25,30)(26,34)(27,29)(28,38)(31,40)(32,37)(33,35)(36,39)Polycyclic presentation:
2^9|a^2=hi,c^2=i,d^2=g,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,d^c=dh,e^c=egi,f^c=fgi,e^d=egi80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 9399550);