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 | #39148 | C2 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
4358 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 40 points:
(9,10)(11,13)(12,20)(14,15)(16,18)(17,21)(19,23)(22,24)(25,26)(27,32)(28,30)(29,33)(31,35)(34,39)(36,40)(37,38);(1,2)(3,4)(5,6)(7,8)(10,15)(11,18)(12,21)(22,23)(25,27,31,39)(26,32,35,34)(28,37,29,36)(30,38,33,40);(2,6)(4,7)(10,15)(13,16)(17,20)(22,23)(25,28,31,29)(26,33,35,30)(27,36,39,37)(32,38,34,40);(9,11)(10,16)(12,19)(13,15)(14,18)(17,23)(20,22)(21,24)(25,29,31,28)(26,33,35,30)(27,37,39,36)(32,38,34,40);(1,3)(2,7)(4,6)(5,8)(9,12,14,21)(10,17,15,20)(11,19,18,24)(13,22,16,23)(25,27)(26,34)(28,36)(29,37)(30,38)(31,39)(32,35)(33,40);(1,4,5,7)(2,3,6,8)(9,13,14,16)(10,11,15,18)(12,22,21,23)(17,19,20,24)(25,30,31,33)(26,28,35,29)(27,38,39,40)(32,37,34,36)Polycyclic presentation:
2^9|b^2=gh,c^2=gh,d^2=gh,e^2=g,f^2=gi,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,e^b=egi,f^c=fgi86 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8138623);