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 | #2428 | C22 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
4351 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 40 points:
(9,10,14,13)(11,17,18,15)(12,20,21,16)(19,22,24,23)(25,26,32,31)(27,37,39,33)(28,40,38,34)(29,36,30,35);(1,2,5,4)(3,7,8,6)(11,18)(15,17)(19,24)(22,23)(25,27)(26,33)(28,29)(30,38)(31,37)(32,39)(34,36)(35,40);(2,4)(6,7)(9,11)(10,15)(12,19)(13,17)(14,18)(16,22)(20,23)(21,24)(25,28,32,38)(26,34,31,40)(27,29,39,30)(33,36,37,35);(1,3)(2,6)(4,7)(5,8)(25,29,32,30)(26,35,31,36)(27,28,39,38)(33,40,37,34);(9,12,14,21)(10,16,13,20)(11,19,18,24)(15,22,17,23)(25,30,32,29)(26,36,31,35)(27,38,39,28)(33,34,37,40);(1,4,5,2)(3,7,8,6)(9,13,14,10)(11,15,18,17)(12,20,21,16)(19,22,24,23)(25,31,32,26)(27,37,39,33)(28,34,38,40)(29,36,30,35)Polycyclic presentation:
2^9|a^2=h,b^2=ghi,c^2=g,d^2=g,e^2=h,f^2=gi,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dghi,f^c=fgi92 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8102923);