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 | |
4686 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 40 points:
(1,2,6,4)(3,8,5,7)(25,26)(27,33)(28,30)(29,32)(31,35)(34,39)(36,37)(38,40);(9,10,14,12)(11,17,19,15)(13,20,21,16)(18,22,24,23)(25,27,30,35)(26,31,28,33)(29,34,37,40)(32,38,36,39);(1,3,6,5)(2,7,4,8)(10,12)(15,17)(16,20)(22,23)(29,37)(32,36)(34,40)(38,39);(1,3,6,5)(2,8,4,7)(9,11,14,19)(10,15,12,17)(13,18,21,24)(16,22,20,23)(25,28,30,26)(27,31,35,33)(29,32,37,36)(34,39,40,38);(1,4,6,2)(3,8,5,7)(9,12,14,10)(11,15,19,17)(13,20,21,16)(18,22,24,23)(25,29)(26,32)(27,34)(28,36)(30,37)(31,38)(33,39)(35,40);(1,5,6,3)(2,8,4,7)(9,13,14,21)(10,16,12,20)(11,18,19,24)(15,22,17,23)(25,27)(26,33)(28,31)(29,34)(30,35)(32,39)(36,38)(37,40)Polycyclic presentation:
2^9|a^2=h,b^2=gi,c^2=h,d^2=ghi,e^2=hi,f^2=hi,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, 10315005);