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 | |
4565 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 40 points:
(1,2,5,4)(3,7,8,6)(9,10)(11,12)(13,17)(14,15)(16,20)(18,19)(21,22)(23,24);(1,3,5,8)(2,6,4,7)(9,10,14,15)(11,12,16,20)(13,17,21,22)(18,19,23,24)(25,26,30,31)(27,29,33,38)(28,32,37,39)(34,36,35,40);(1,2,5,4)(3,6,8,7)(10,15)(11,16)(13,21)(19,24)(26,31)(29,38)(32,39)(36,40);(9,11)(10,12)(13,18)(14,16)(15,20)(17,19)(21,23)(22,24)(25,27,30,33)(26,29,31,38)(28,35,37,34)(32,40,39,36);(1,2,5,4)(3,6,8,7)(9,12)(10,16)(11,15)(13,19)(14,20)(17,23)(18,22)(21,24)(25,28,30,37)(26,32,31,39)(27,34,33,35)(29,36,38,40);(1,4,5,2)(3,7,8,6)(9,13)(10,17)(11,18)(12,19)(14,21)(15,22)(16,23)(20,24)(25,29)(26,33)(27,31)(28,36)(30,38)(32,35)(34,39)(37,40)Polycyclic presentation:
2^9|a^2=gh,b^2=ghi,c^2=gh,d^2=hi,e^2=gi,f^2=gh,b^a=bg,c^a=ch,e^a=eh,c^b=ci,d^c=dh,f^c=fi,e^d=ei98 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7685164);