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 | |
4354 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)(3,4)(5,8)(6,7)(9,10)(11,17)(12,16)(13,18)(14,15)(19,24)(20,21)(22,23)(25,26)(27,28)(29,37)(30,31)(32,35)(33,39)(34,38)(36,40);(1,3,5,7)(2,4,8,6)(9,11)(10,13)(12,20)(14,18)(15,17)(16,24)(19,22)(21,23)(26,31)(28,32)(33,37)(36,38);(1,3)(2,6)(4,8)(5,7)(10,15)(11,18)(16,23)(19,20)(25,27,30,35)(26,32,31,28)(29,34,39,40)(33,38,37,36);(1,2,5,8)(3,4,7,6)(9,11)(10,13)(12,19)(14,18)(15,17)(16,21)(20,22)(23,24)(25,27)(26,28)(29,34)(30,35)(31,32)(33,36)(37,38)(39,40);(1,3)(2,4)(5,7)(6,8)(9,12,14,22)(10,16,15,23)(11,19,18,20)(13,21,17,24)(25,28)(26,27)(29,38)(30,32)(31,35)(33,40)(34,37)(36,39);(1,4)(2,7)(3,8)(5,6)(9,13)(10,11)(12,21)(14,17)(15,18)(16,19)(20,23)(22,24)(25,29,30,39)(26,33,31,37)(27,34,35,40)(28,36,32,38)Polycyclic presentation:
2^9|b^2=gh,c^2=hi,d^2=gh,e^2=ghi,f^2=hi,b^a=bg,c^a=ch,d^a=di,f^a=fh,c^b=ci,e^b=eh,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, 9633730);