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 |
| prime radical | #48100 | C2 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
4465 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 40 points:
(2,5)(6,8)(9,10)(11,15)(12,19)(13,21)(14,17)(16,22)(18,20)(23,24)(25,26,31,32)(27,36,37,33)(28,40,34,29)(30,35,38,39);(3,7)(6,8)(9,11,14,19)(10,15,17,12)(13,18,22,24)(16,23,21,20)(26,32)(27,37)(28,34)(35,39);(1,2)(3,6)(4,5)(7,8)(9,10,14,17)(11,12,19,15)(13,16,22,21)(18,20,24,23)(25,27)(26,33)(28,38)(29,39)(30,34)(31,37)(32,36)(35,40);(1,3)(2,6)(4,7)(5,8)(9,10)(11,12)(13,16)(14,17)(15,19)(18,20)(21,22)(23,24)(25,28,31,34)(26,29,32,40)(27,38,37,30)(33,39,36,35);(1,2)(3,6)(4,5)(7,8)(9,12)(10,11)(13,20)(14,15)(16,18)(17,19)(21,24)(22,23)(25,29)(26,34)(27,39)(28,32)(30,33)(31,40)(35,37)(36,38);(9,13)(10,16)(11,18)(12,20)(14,22)(15,23)(17,21)(19,24)(25,30,31,38)(26,35,32,39)(27,34,37,28)(29,33,40,36)Polycyclic presentation:
2^9|a^2=g,b^2=gi,c^2=gi,d^2=g,f^2=g,b^a=bg,c^a=ch,d^a=dg,e^a=eh,f^a=fgi,c^b=ci,d^b=dh,d^c=dgi,e^c=egi86 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8034388);