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 | |
4480 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 40 points:
(1,2,5,3)(4,7,8,6)(9,10)(11,16)(12,13)(14,22)(15,17)(18,23)(19,21)(20,24);(2,3)(6,7)(9,11,15,12)(10,16,17,13)(14,19,23,20)(18,24,22,21)(25,26,30,31)(27,28,32,36)(29,33,37,38)(34,35,39,40);(1,3,5,2)(4,7,8,6)(10,17)(11,12)(14,23)(21,24)(26,31)(27,32)(29,37)(35,40);(1,4)(2,6)(3,7)(5,8)(9,12,15,11)(10,13,17,16)(14,20,23,19)(18,21,22,24)(25,27)(26,28)(29,34)(30,32)(31,36)(33,35)(37,39)(38,40);(9,13)(10,12)(11,17)(14,21)(15,16)(18,20)(19,22)(23,24)(25,28)(26,32)(27,31)(29,35)(30,36)(33,39)(34,38)(37,40);(9,14)(10,18)(11,19)(12,20)(13,21)(15,23)(16,24)(17,22)(25,29)(26,33)(27,34)(28,35)(30,37)(31,38)(32,39)(36,40)Polycyclic presentation:
2^9|a^2=g,b^2=gi,c^2=g,d^2=h,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,d^c=dgi,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, 8815623);