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 | |
4505 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 40 points:
(2,4)(6,8)(9,10,14,15)(11,17,18,12)(13,21,22,16)(19,20,24,23);(3,7)(6,8)(10,15)(11,18)(16,21)(19,24)(25,26)(27,33)(28,36)(29,30)(31,35)(32,37)(34,39)(38,40);(1,2,5,4)(3,6,7,8)(10,15)(12,17)(13,22)(19,24)(26,30)(28,37)(31,33)(34,38);(9,11,14,18)(10,12,15,17)(13,19,22,24)(16,20,21,23)(25,27)(26,31)(28,38)(29,35)(30,33)(32,40)(34,37)(36,39);(1,3,5,7)(2,6,4,8)(9,12)(10,11)(13,20)(14,17)(15,18)(16,19)(21,24)(22,23)(25,28)(26,32)(27,34)(29,37)(30,36)(31,39)(33,40)(35,38);(1,4,5,2)(3,8,7,6)(9,13,14,22)(10,16,15,21)(11,19,18,24)(12,20,17,23)(25,26)(27,31)(28,32)(29,30)(33,35)(34,39)(36,37)(38,40)Polycyclic presentation:
2^9|a^2=g,c^2=gh,d^2=g,e^2=gh,f^2=h,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,d^b=dgi,e^b=eghi,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, 8948136);