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 | |
4446 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 40 points:
(9,10)(11,17)(12,16)(13,18)(14,15)(19,24)(20,21)(22,23)(26,31)(27,34)(33,40)(35,36);(1,2)(3,6)(4,7)(5,8)(9,11)(10,13)(12,20)(14,18)(15,17)(16,24)(19,22)(21,23)(25,26)(27,32)(28,33)(29,34)(30,31)(35,37)(36,39)(38,40);(1,3)(2,5)(4,8)(6,7)(10,15)(11,18)(16,23)(19,20);(1,2)(3,5)(4,7)(6,8)(9,11)(10,13)(12,19)(14,18)(15,17)(16,21)(20,22)(23,24)(25,27,30,34)(26,32,31,29)(28,36,38,35)(33,39,40,37);(1,3)(2,6)(4,8)(5,7)(9,12,14,22)(10,16,15,23)(11,19,18,20)(13,21,17,24)(25,28,30,38)(26,33,31,40)(27,35,34,36)(29,39,32,37);(1,2)(3,5)(4,7)(6,8)(9,13)(10,11)(12,21)(14,17)(15,18)(16,19)(20,23)(22,24)(25,29,30,32)(26,34,31,27)(28,37,38,39)(33,35,40,36)Polycyclic presentation:
2^9|d^2=gh,e^2=g,f^2=gh,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,d^c=dh,e^d=egi,f^e=fgi86 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 9996175);