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 | |
4429 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 40 points:
(1,2,5,3)(4,7,8,6)(9,10)(11,17)(12,13)(14,18)(15,21)(16,22)(19,23)(20,24)(25,26,30,33)(27,34,29,31)(28,37,38,32)(35,39,36,40);(2,3)(6,7)(9,11)(10,15)(12,19)(13,20)(14,21)(16,23)(17,18)(22,24)(25,27,30,29)(26,31,33,34)(28,35,38,36)(32,39,37,40);(1,3,5,2)(4,7,8,6)(25,28)(26,32)(27,35)(29,36)(30,38)(31,39)(33,37)(34,40);(1,4,5,8)(2,6,3,7)(9,12,14,22)(10,16,18,13)(11,19,21,24)(15,23,17,20)(25,28)(26,32)(27,36)(29,35)(30,38)(31,40)(33,37)(34,39);(9,11)(10,17)(12,19)(13,23)(14,21)(15,18)(16,20)(22,24)(25,26)(27,31)(28,32)(29,34)(30,33)(35,39)(36,40)(37,38);(9,13,14,16)(10,12,18,22)(11,20,21,23)(15,19,17,24)(25,29,30,27)(26,31,33,34)(28,36,38,35)(32,39,37,40)Polycyclic presentation:
2^9|a^2=hi,b^2=h,c^2=i,d^2=gh,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, 8586175);