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 | #37826 | C2 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
4379 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)(3,7)(4,5)(6,8)(9,10,14,17)(11,12,19,15)(13,16,22,24)(18,20,23,21)(25,26,31,36)(27,34,29,32)(28,33,35,30)(37,38,39,40);(2,5)(6,7)(9,11,14,19)(10,15,17,12)(13,18,22,23)(16,21,24,20)(27,29)(32,34)(37,39)(38,40);(1,2)(3,6)(4,5)(7,8)(25,27,31,29)(26,32,36,34)(28,37,35,39)(30,38,33,40);(1,3)(2,6)(4,8)(5,7)(9,12)(10,11)(13,21)(14,15)(16,23)(17,19)(18,24)(20,22)(25,28)(26,33)(27,37)(29,39)(30,36)(31,35)(32,40)(34,38);(9,11)(10,15)(12,17)(13,18)(14,19)(16,21)(20,24)(22,23)(25,29,31,27)(26,34,36,32)(28,39,35,37)(30,40,33,38);(9,13)(10,16)(11,18)(12,20)(14,22)(15,21)(17,24)(19,23)(25,30,31,33)(26,35,36,28)(27,38,29,40)(32,39,34,37)Polycyclic presentation:
2^9|a^2=gi,b^2=ghi,c^2=h,e^2=h,f^2=h,b^a=bg,c^a=ch,d^a=dg,e^a=egi,f^a=fh,c^b=ci,e^b=egi,e^d=eghi,f^d=fgi83 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 10329528);