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 | |
4389 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)(11,15)(12,19)(13,18)(14,24)(16,21)(17,23)(20,22)(26,30)(28,35)(31,36)(34,39);(2,6)(3,8)(9,11)(10,17)(12,22)(13,14)(15,21)(16,23)(18,20)(19,24)(27,33)(31,36)(34,39)(38,40);(1,3)(2,4)(5,8)(6,7)(9,12,16,13)(10,18,21,19)(11,22,23,14)(15,24,17,20)(25,26)(27,31)(28,32)(29,30)(33,36)(34,38)(35,37)(39,40);(1,3)(2,4)(5,8)(6,7)(9,13,16,12)(10,19,21,18)(11,22,23,14)(15,24,17,20)(25,27)(26,31)(28,34)(29,33)(30,36)(32,38)(35,39)(37,40);(1,4)(2,3)(5,7)(6,8)(9,14,16,22)(10,20,21,24)(11,13,23,12)(15,19,17,18)(25,28,29,35)(26,32,30,37)(27,34,33,39)(31,38,36,40);(9,15,16,17)(10,11,21,23)(12,20,13,24)(14,19,22,18)Polycyclic presentation:
2^9|c^2=gi,d^2=gi,e^2=hi,f^2=gi,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,f^c=fgi,e^d=egi86 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8253269);