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 | |
4467 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)(5,7)(6,8)(9,10)(11,19)(12,21)(13,18)(14,22)(15,17)(16,20)(23,24)(25,26)(27,30)(28,35)(29,32)(31,36)(33,37)(34,40)(38,39);(1,3)(2,6)(4,7)(5,8)(9,11,15,20)(10,16,17,19)(12,13,22,23)(14,18,21,24)(27,33)(30,37)(34,39)(38,40);(10,17)(11,20)(12,22)(18,24)(25,26,29,32)(27,30,33,37)(28,31,36,35)(34,38,39,40);(9,12)(10,14)(11,13)(15,22)(16,18)(17,21)(19,24)(20,23)(25,26)(27,30)(28,31)(29,32)(33,37)(34,38)(35,36)(39,40);(1,4,5,6)(2,3,7,8)(9,13)(10,18)(11,12)(14,16)(15,23)(17,24)(19,21)(20,22)(25,27,29,33)(26,30,32,37)(28,34,36,39)(31,38,35,40);(1,3)(2,4)(5,8)(6,7)(9,14)(10,12)(11,18)(13,16)(15,21)(17,22)(19,23)(20,24)(25,28,29,36)(26,31,32,35)(27,34,33,39)(30,38,37,40)Polycyclic presentation:
2^9|b^2=i,c^2=hi,e^2=gh,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.
| maximal normal subgroup of | #72219279, #72219281 |
| maximal quotient of | #72219272 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 9637538);