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 | |
4605 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 40 points:
(2,4)(6,7)(9,10)(11,18)(12,20)(13,17)(14,15)(16,23)(19,22)(21,24)(25,26,31,34)(27,32,35,30)(28,29,33,39)(36,37,40,38);(1,2,5,4)(3,7,8,6)(10,15)(11,17)(16,20)(19,21)(25,27,31,35)(26,32,34,30)(28,36,33,40)(29,37,39,38);(1,3,5,8)(2,6,4,7)(10,15)(13,18)(16,20)(22,24)(27,35)(30,32)(36,40)(37,38);(1,4,5,2)(3,7,8,6)(9,11,14,17)(10,13,15,18)(12,21,23,19)(16,24,20,22)(25,28)(26,29)(27,36)(30,38)(31,33)(32,37)(34,39)(35,40);(9,12,14,23)(10,16,15,20)(11,19,17,21)(13,22,18,24)(25,29)(26,33)(27,37)(28,34)(30,36)(31,39)(32,40)(35,38);(9,13)(10,17)(11,15)(12,22)(14,18)(16,21)(19,20)(23,24)(25,30)(26,27)(28,38)(29,36)(31,32)(33,37)(34,35)(39,40)Polycyclic presentation:
2^9|a^2=ghi,b^2=i,c^2=gh,d^2=g,e^2=h,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,f^c=fgi,e^d=egi86 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8192472);