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 | |
4170 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 40 points:
(9,10,14,12)(11,17,19,15)(13,16,21,20)(18,23,24,22)(25,26)(27,32)(28,34)(29,35)(30,37)(31,33)(36,40)(38,39);(1,2,4,6)(3,5,8,7)(10,12)(15,17)(16,20)(22,23)(25,27,31,28)(26,32,33,34)(29,37,39,36)(30,38,40,35);(1,2)(3,5)(4,6)(7,8)(26,33)(27,28)(35,38)(36,37);(9,11,14,19)(10,15,12,17)(13,18,21,24)(16,22,20,23)(25,28,31,27)(26,34,33,32)(29,36,39,37)(30,35,40,38);(1,2)(3,7)(4,6)(5,8)(9,12,14,10)(11,17,19,15)(13,16,21,20)(18,22,24,23)(25,29)(26,35)(27,37)(28,36)(30,32)(31,39)(33,38)(34,40);(1,3)(2,5)(4,8)(6,7)(9,13)(10,16)(11,18)(12,20)(14,21)(15,22)(17,23)(19,24)(25,30,31,40)(26,36,33,37)(27,38,28,35)(29,34,39,32)Polycyclic presentation:
2^9|a^2=g,b^2=i,d^2=gh,e^2=g,f^2=h,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eghi,d^c=dh,f^e=fgi83 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 9696239);