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 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
4447 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 40 points:
(2,4)(6,8)(9,10,14,11)(12,15,19,17)(13,16,21,18)(20,22,24,23)(25,26)(27,32)(28,38)(29,33)(30,34)(31,35)(36,40)(37,39);(3,7)(6,8)(10,11)(12,19)(16,18)(20,24)(25,27)(26,32)(28,30)(29,39)(31,36)(33,37)(34,38)(35,40);(1,2,5,4)(3,6,7,8)(13,21)(16,18)(20,24)(22,23)(27,36)(30,37)(32,40)(34,39);(1,3,5,7)(2,6,4,8)(9,11,14,10)(12,15,19,17)(13,18,21,16)(20,22,24,23)(25,28)(26,29)(27,37)(30,36)(31,33)(32,34)(35,38)(39,40);(1,4,5,2)(3,8,7,6)(9,12)(10,15)(11,17)(13,20)(14,19)(16,22)(18,23)(21,24)(25,29)(26,33)(27,34)(28,35)(30,32)(31,38)(36,39)(37,40);(9,13,14,21)(10,16,11,18)(12,20,19,24)(15,22,17,23)(25,30,31,37)(26,34,35,39)(27,28,36,33)(29,32,38,40)Polycyclic presentation:
2^9|a^2=g,c^2=h,d^2=gh,e^2=h,f^2=gi,b^a=bg,c^a=ch,d^a=di,e^a=eh,c^b=ci,d^b=dghi,e^b=egi,f^c=fgi,e^d=egi80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8325401);