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 | |
4966 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 6 generators on 40 points:
(1,2,5,7)(3,4,8,6)(9,10,14,12)(11,15,17,19)(13,16,22,20)(18,23,21,24)(25,26)(27,32)(28,30)(29,36)(31,35)(33,38)(34,39)(37,40);(1,3,5,8)(2,6,7,4)(9,11)(10,15)(12,19)(13,21)(14,17)(16,24)(18,22)(20,23)(25,27)(26,31)(28,32)(29,37)(30,35)(33,40)(34,38)(36,39);(1,3)(2,4)(5,8)(6,7)(11,17)(15,19)(18,21)(23,24)(25,26,30,28)(27,31,35,32)(29,33,38,36)(34,39,37,40);(1,4)(2,3)(5,6)(7,8)(9,12,14,10)(11,15,17,19)(13,20,22,16)(18,23,21,24)(25,28,30,26)(27,31,35,32)(29,36,38,33)(34,39,37,40);(9,11)(10,15)(12,19)(13,18)(14,17)(16,23)(20,24)(21,22)(25,27)(26,32)(28,31)(29,34)(30,35)(33,40)(36,39)(37,38);(9,13)(10,16)(11,18)(12,20)(14,22)(15,23)(17,21)(19,24)(25,29)(26,33)(27,34)(28,36)(30,38)(31,39)(32,40)(35,37)Polycyclic presentation:
2^9|a^2=i,b^2=gh,c^2=h,d^2=gi,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,d^b=dgi,f^b=fgi,e^c=egi,e^d=eghi83 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 10259626);