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 | |
4618 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 40 points:
(1,2,4,6)(3,5,7,8)(9,10,14,19)(11,16,20,15)(12,17,23,22)(13,18,24,21)(25,26)(27,34)(28,33)(29,38)(30,31)(32,37)(35,36)(39,40);(1,3,4,7)(2,5,6,8)(11,20)(12,23)(15,16)(17,22)(25,27)(26,32)(28,35)(29,36)(30,34)(31,37)(33,39)(38,40);(1,2)(3,5)(4,6)(7,8)(9,11)(10,15)(12,21)(13,22)(14,20)(16,19)(17,24)(18,23)(25,28,31,38)(26,33,30,29)(27,35,37,40)(32,39,34,36);(1,3)(2,5)(4,7)(6,8)(9,11)(10,16)(12,21)(13,17)(14,20)(15,19)(18,23)(22,24)(25,29)(26,28)(27,36)(30,38)(31,33)(32,35)(34,40)(37,39);(1,2)(3,5)(4,6)(7,8)(9,12)(10,17)(11,21)(13,16)(14,23)(15,24)(18,20)(19,22)(25,30,31,26)(27,32,37,34)(28,29,38,33)(35,39,40,36);(9,13)(10,18)(11,22)(12,16)(14,24)(15,23)(17,20)(19,21)(25,27)(26,34)(28,35)(29,39)(30,32)(31,37)(33,36)(38,40)Polycyclic presentation:
2^9|a^2=h,b^2=hi,c^2=g,e^2=g,b^a=bg,c^a=ch,d^a=dg,e^a=eghi,c^b=ci,d^b=dh,e^b=egi,f^d=fgi83 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8382575);