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 | |
4590 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 40 points:
(1,2,4,5)(3,7,8,6)(9,10)(11,16)(12,18)(13,17)(14,15)(19,23)(20,24)(21,22)(25,26,31,28)(27,32,35,29)(30,33,38,36)(34,39,40,37);(2,5)(6,7)(10,15)(16,18)(17,22)(23,24)(25,27)(26,32)(28,29)(30,34)(31,35)(33,39)(36,37)(38,40);(1,2)(3,6)(4,5)(7,8)(9,11,14,12)(10,16,15,18)(13,19,21,20)(17,23,22,24)(25,26,31,28)(27,29,35,32)(30,33,38,36)(34,37,40,39);(1,2)(3,6)(4,5)(7,8)(9,12,14,11)(10,16,15,18)(13,19,21,20)(17,24,22,23)(25,28,31,26)(27,29,35,32)(30,33,38,36)(34,39,40,37);(1,3)(2,6)(4,8)(5,7)(9,10)(11,18)(12,16)(13,17)(14,15)(19,24)(20,23)(21,22)(25,29)(26,27)(28,35)(30,37)(31,32)(33,34)(36,40)(38,39);(9,13)(10,17)(11,19)(12,20)(14,21)(15,22)(16,23)(18,24)(25,30,31,38)(26,33,28,36)(27,34,35,40)(29,37,32,39)Polycyclic presentation:
2^9|a^2=i,c^2=gi,d^2=gi,f^2=hi,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,e^b=egi,e^c=egi,f^d=fgi89 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8556263);