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 | #54822 | C2 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
4386 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 40 points:
(1,2,5,6)(3,4,7,8)(9,10,13,15)(11,17,19,14)(12,16,21,23)(18,24,20,22)(25,26,31,30)(27,32,34,29)(28,33,37,38)(35,39,40,36);(2,6)(3,7)(9,11,13,19)(10,14,15,17)(12,18,21,20)(16,22,23,24)(25,27,31,34)(26,32,30,29)(28,35,37,40)(33,39,38,36);(1,3)(2,4)(5,7)(6,8)(10,15)(14,17)(16,23)(22,24)(27,34)(29,32)(35,40)(36,39);(1,4)(2,3)(5,8)(6,7)(9,11)(10,14)(12,20)(13,19)(15,17)(16,24)(18,21)(22,23)(25,28)(26,33)(27,35)(29,36)(30,38)(31,37)(32,39)(34,40);(9,10)(11,14)(12,16)(13,15)(17,19)(18,22)(20,24)(21,23)(25,29)(26,27)(28,36)(30,34)(31,32)(33,35)(37,39)(38,40);(9,12)(10,16)(11,18)(13,21)(14,22)(15,23)(17,24)(19,20)(25,30,31,26)(27,29,34,32)(28,33,37,38)(35,39,40,36)Polycyclic presentation:
2^9|a^2=hi,b^2=gi,f^2=ghi,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,e^c=egi,f^d=fgi83 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8458423);