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 | #44770 | C2 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
4262 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 40 points:
(1,2,5,8)(3,6,4,7)(9,10,15,12)(11,13,18,16)(14,17,22,20)(19,21,24,23)(25,26)(27,33)(28,39)(29,30)(31,34)(32,35)(36,37)(38,40);(1,2)(3,6)(4,7)(5,8)(9,11)(10,16)(12,13)(14,19)(15,18)(17,23)(20,21)(22,24)(25,27,32,38)(26,33,35,40)(28,31,29,36)(30,37,39,34);(11,18)(13,16)(19,24)(21,23)(25,28,32,29)(26,30,35,39)(27,36,38,31)(33,34,40,37);(9,12,15,10)(11,16,18,13)(14,17,22,20)(19,21,24,23)(25,29,32,28)(26,30,35,39)(27,31,38,36)(33,34,40,37);(1,3,5,4)(2,6,8,7)(9,13,15,16)(10,11,12,18)(14,21,22,23)(17,19,20,24)(25,30)(26,28)(27,37)(29,35)(31,33)(32,39)(34,38)(36,40);(1,4,5,3)(2,7,8,6)(9,14,15,22)(10,17,12,20)(11,19,18,24)(13,21,16,23)(25,31)(26,34)(27,29)(28,38)(30,33)(32,36)(35,37)(39,40)Polycyclic presentation:
2^9|a^2=g,b^2=h,c^2=h,d^2=i,e^2=g,f^2=g,b^a=bg,c^a=ch,e^a=ei,c^b=ci,d^b=di,e^c=ei,f^c=fh,e^d=ehi,f^d=fi95 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7995379);