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 | |
4415 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 40 points:
(1,2)(3,7)(4,8)(5,6)(9,10)(11,12)(13,17)(14,15)(16,19)(18,20)(21,22)(23,24)(25,26)(27,38)(28,34)(29,33)(30,35)(31,36)(32,37)(39,40);(2,6)(3,8)(10,15)(12,16)(17,24)(20,21)(25,27,32,29)(26,33,37,38)(28,36,39,30)(31,34,35,40);(1,3)(2,4)(5,8)(6,7)(9,11)(10,16)(12,15)(13,18)(14,19)(17,21)(20,24)(22,23)(25,28,32,39)(26,34,37,40)(27,36,29,30)(31,33,35,38);(1,3)(2,4)(5,8)(6,7)(9,12,14,16)(10,11,15,19)(13,21,23,20)(17,22,24,18)(25,29,32,27)(26,33,37,38)(28,30,39,36)(31,34,35,40);(1,4)(2,3)(5,7)(6,8)(9,11)(10,16)(12,15)(13,22)(14,19)(17,20)(18,23)(21,24)(25,30)(26,35)(27,28)(29,39)(31,37)(32,36)(33,40)(34,38);(9,13,14,23)(10,17,15,24)(11,18,19,22)(12,20,16,21)(25,31,32,35)(26,36,37,30)(27,34,29,40)(28,33,39,38)Polycyclic presentation:
2^9|b^2=gh,c^2=gh,d^2=gi,f^2=gi,b^a=bg,c^a=ch,d^a=di,e^a=eh,c^b=ci,d^b=dh,f^d=fgi,f^e=fgi83 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8595501);