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 |
| prime radical | #45789 | C2 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
4360 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 40 points:
(2,6)(7,8)(9,10)(11,12)(13,21)(14,18)(15,16)(17,23)(19,24)(20,22)(25,26)(27,33)(28,36)(29,35)(30,32)(31,38)(34,37)(39,40);(1,2)(3,7)(4,8)(5,6)(9,11,15,14)(10,12,16,18)(13,19,23,22)(17,20,21,24)(25,27)(26,29)(28,34)(30,35)(31,37)(32,33)(36,39)(38,40);(10,16)(12,18)(17,21)(20,24)(25,26,30,32)(27,33,35,29)(28,36,38,31)(34,37,40,39);(1,3,5,4)(2,7,6,8)(9,12,15,18)(10,11,16,14)(13,20,23,24)(17,19,21,22)(25,26)(27,29)(28,31)(30,32)(33,35)(34,37)(36,38)(39,40);(1,3,5,4)(2,7,6,8)(9,13,15,23)(10,17,16,21)(11,19,14,22)(12,20,18,24)(25,28,30,38)(26,31,32,36)(27,34,35,40)(29,37,33,39);(1,4,5,3)(2,8,6,7)(9,14,15,11)(10,18,16,12)(13,22,23,19)(17,24,21,20)(25,29)(26,27)(28,37)(30,33)(31,34)(32,35)(36,40)(38,39)Polycyclic presentation:
2^9|b^2=hi,c^2=i,d^2=gh,e^2=ghi,f^2=gh,b^a=bg,c^a=ch,e^a=eh,f^a=fi,c^b=ci,d^c=dh,e^c=ei104 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7618843);