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 | |
4600 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)(12,13)(14,18)(15,21)(16,22)(19,23)(20,24)(25,26,30,34)(27,35,37,31)(28,38,39,32)(29,33,36,40);(1,2)(3,6)(4,7)(5,8)(9,11)(10,15)(12,19)(13,20)(14,21)(16,23)(17,18)(22,24)(25,27,30,37)(26,31,34,35)(28,29,39,36)(32,33,38,40);(1,3)(2,5)(4,8)(6,7)(25,28,30,39)(26,32,34,38)(27,36,37,29)(31,40,35,33);(1,2)(3,5)(4,7)(6,8)(9,12,14,22)(10,16,18,13)(11,19,21,24)(15,23,17,20)(25,28)(26,32)(27,29)(30,39)(31,33)(34,38)(35,40)(36,37);(1,3)(2,6)(4,8)(5,7)(9,13,14,16)(10,12,18,22)(11,20,21,23)(15,19,17,24)(25,29)(26,33)(27,28)(30,36)(31,32)(34,40)(35,38)(37,39);(1,2)(3,5)(4,7)(6,8)(9,11)(10,17)(12,19)(13,23)(14,21)(15,18)(16,20)(22,24)(25,26)(27,31)(28,32)(29,33)(30,34)(35,37)(36,40)(38,39)Polycyclic presentation:
2^9|a^2=h,b^2=h,c^2=h,d^2=gh,e^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, 9995576);