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 | |
4617 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)(9,10,14,18)(11,19,22,15)(12,23,21,16)(13,17,20,24);(2,3)(6,7)(9,11)(10,15)(12,20)(13,21)(14,22)(16,24)(17,23)(18,19)(25,26)(27,32)(28,31)(29,33)(30,34)(35,36)(37,39)(38,40);(1,3,5,2)(4,7,8,6)(9,10)(11,15)(12,16)(13,17)(14,18)(19,22)(20,24)(21,23)(25,27)(26,29)(28,35)(30,33)(31,38)(32,34)(36,39)(37,40);(1,4,5,8)(2,6,3,7)(9,10)(11,19)(12,16)(13,24)(14,18)(15,22)(17,20)(21,23)(25,28,30,37)(26,31,34,39)(27,35,33,40)(29,38,32,36);(9,12)(10,16)(11,20)(13,22)(14,21)(15,24)(17,19)(18,23)(25,27)(26,32)(28,35)(29,34)(30,33)(31,36)(37,40)(38,39);(9,13,14,20)(10,17,18,24)(11,21,22,12)(15,23,19,16)(25,29,30,32)(26,33,34,27)(28,36,37,38)(31,35,39,40)Polycyclic presentation:
2^9|a^2=g,c^2=gh,d^2=i,f^2=gi,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,f^d=fgi,f^e=fgi89 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8587190);