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 | |
4582 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 40 points:
(2,6)(3,8)(9,10)(11,17)(12,16)(13,18)(14,15)(19,24)(20,21)(22,23)(25,26,30,35)(27,31,38,32)(28,33,34,29)(36,37,39,40);(1,2)(3,7)(4,8)(5,6)(9,11)(10,13)(12,20)(14,18)(15,17)(16,24)(19,22)(21,23)(25,27)(26,31)(28,39)(29,37)(30,38)(32,35)(33,40)(34,36);(1,3,5,8)(2,7,6,4)(10,15)(11,18)(16,23)(19,20)(25,27,30,38)(26,32,35,31)(28,36,34,39)(29,37,33,40);(1,4)(2,8)(3,6)(5,7)(9,11)(10,13)(12,19)(14,18)(15,17)(16,21)(20,22)(23,24)(25,27)(26,32)(28,36)(29,37)(30,38)(31,35)(33,40)(34,39);(1,4)(2,3)(5,7)(6,8)(9,12)(10,16)(11,19)(13,21)(14,22)(15,23)(17,24)(18,20)(25,28)(26,33)(27,36)(29,35)(30,34)(31,37)(32,40)(38,39);(1,2)(3,7)(4,8)(5,6)(9,13)(10,11)(12,21)(14,17)(15,18)(16,19)(20,23)(22,24)(25,29,30,33)(26,34,35,28)(27,37,38,40)(31,36,32,39)Polycyclic presentation:
2^9|a^2=gh,c^2=gi,f^2=gh,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, 9627719);