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 | |
4593 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 40 points:
(1,2,5,6)(3,4,7,8)(9,10,14,15)(11,17,18,13)(12,16,20,23)(19,24,21,22)(25,26)(27,36)(28,33)(29,30)(31,35)(32,37)(34,40)(38,39);(2,6)(3,7)(10,15)(11,18)(12,20)(22,24)(27,37)(28,39)(32,36)(33,38);(1,3)(2,4)(5,7)(6,8)(11,18)(12,20)(13,17)(16,23)(25,27,31,37)(26,32,35,36)(28,29,39,34)(30,33,40,38);(1,4)(2,3)(5,8)(6,7)(9,11)(10,13)(12,21)(14,18)(15,17)(16,24)(19,20)(22,23)(25,28,31,39)(26,33,35,38)(27,34,37,29)(30,36,40,32);(9,12)(10,16)(11,19)(13,22)(14,20)(15,23)(17,24)(18,21)(25,29,31,34)(26,30,35,40)(27,28,37,39)(32,38,36,33);(9,13)(10,11)(12,22)(14,17)(15,18)(16,19)(20,24)(21,23)(25,30)(26,34)(27,38)(28,36)(29,35)(31,40)(32,39)(33,37)Polycyclic presentation:
2^9|a^2=g,c^2=h,d^2=h,e^2=h,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,d^b=dgi,e^b=eghi,d^c=dgi,e^c=egi,f^c=fgi,e^d=egi92 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8966469);