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 | |
4670 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)(9,10,14,15)(11,16,19,23)(12,17,21,20)(13,18,24,22)(25,26,30,31)(27,28,33,32)(29,37,38,34)(35,39,40,36);(3,7)(5,8)(10,15)(12,21)(16,23)(18,22)(25,27)(26,28)(29,35)(30,33)(31,32)(34,36)(37,39)(38,40);(1,2)(3,5)(4,6)(7,8)(11,19)(13,24)(16,23)(18,22)(26,31)(27,33)(29,38)(36,39);(9,11,14,19)(10,16,15,23)(12,22,21,18)(13,20,24,17)(25,27)(26,32)(28,31)(29,35)(30,33)(34,39)(36,37)(38,40);(1,3)(2,5)(4,7)(6,8)(9,12)(10,17)(11,18)(13,23)(14,21)(15,20)(16,24)(19,22)(25,28,30,32)(26,33,31,27)(29,36,38,39)(34,40,37,35);(1,2)(3,5)(4,6)(7,8)(9,13,14,24)(10,18,15,22)(11,20,19,17)(12,23,21,16)(25,29)(26,34)(27,35)(28,36)(30,38)(31,37)(32,39)(33,40)Polycyclic presentation:
2^9|a^2=gh,d^2=g,e^2=i,f^2=g,b^a=bg,c^a=ch,d^a=di,f^a=fh,c^b=ci,e^b=eghi,d^c=dgi,f^c=fgi,e^d=egi80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8837893);