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 | |
4662 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 40 points:
(1,2,5,3)(4,7,8,6)(9,10,15,16)(11,13,12,19)(14,17,23,24)(18,21,20,22)(25,26)(27,31)(28,37)(29,33)(30,34)(32,36)(35,40)(38,39);(2,3)(6,7)(9,11)(10,13)(12,15)(14,18)(16,19)(17,21)(20,23)(22,24)(25,27,30,35)(26,31,34,40)(28,29,36,39)(32,38,37,33);(1,3,5,2)(4,7,8,6)(10,16)(11,12)(17,24)(18,20)(27,35)(28,36)(31,40)(32,37);(1,4,5,8)(2,6,3,7)(9,12,15,11)(10,13,16,19)(14,18,23,20)(17,22,24,21)(25,28)(26,32)(27,29)(30,36)(31,38)(33,40)(34,37)(35,39);(9,13)(10,11)(12,16)(14,22)(15,19)(17,20)(18,24)(21,23)(25,26)(27,31)(28,32)(29,38)(30,34)(33,39)(35,40)(36,37);(9,14)(10,17)(11,18)(12,20)(13,21)(15,23)(16,24)(19,22)(25,29)(26,33)(27,36)(28,35)(30,39)(31,32)(34,38)(37,40)Polycyclic presentation:
2^9|a^2=gh,b^2=ghi,c^2=g,d^2=gh,b^a=bg,c^a=ch,d^a=di,e^a=eh,c^b=ci,d^b=dh,d^c=dgi,f^d=fgi,f^e=fgi83 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8641557);