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 | #34547 | C2 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
4405 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 40 points:
(1,2)(3,7)(4,6)(5,8)(9,10,14,17)(11,15,19,12)(13,21,22,16)(18,20,24,23)(25,26,31,36)(27,28,33,32)(29,35,30,34)(37,40,38,39);(1,2,4,6)(3,5,8,7)(9,11)(10,15)(12,17)(13,18)(14,19)(16,23)(20,21)(22,24)(27,33)(28,32)(37,38)(39,40);(1,2)(3,5)(4,6)(7,8)(9,12)(10,11)(13,20)(14,15)(16,18)(17,19)(21,24)(22,23)(25,27)(26,32)(28,36)(29,37)(30,38)(31,33)(34,40)(35,39);(1,3)(2,5)(4,8)(6,7)(9,11)(10,15)(12,17)(13,18)(14,19)(16,23)(20,21)(22,24)(25,28,31,32)(26,33,36,27)(29,39,30,40)(34,38,35,37);(25,29,31,30)(26,34,36,35)(27,37,33,38)(28,39,32,40);(9,13,14,22)(10,16,17,21)(11,18,19,24)(12,20,15,23)(25,30,31,29)(26,35,36,34)(27,38,33,37)(28,40,32,39)Polycyclic presentation:
2^9|a^2=gi,b^2=g,d^2=h,e^2=h,f^2=gi,b^a=bg,c^a=ch,d^a=dg,e^a=eh,f^a=fgi,c^b=ci,d^b=dh,d^c=dgi92 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8015222);