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 |
| prime radical | #50006 | C2 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
4382 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 40 points:
(9,10,14,16)(11,15,18,12)(13,21,24,17)(19,20,22,23)(25,26)(27,29)(28,32)(30,33)(31,34)(35,37)(36,39)(38,40);(1,2,5,6)(3,7,8,4)(9,11)(10,15)(12,16)(13,22)(14,18)(17,20)(19,24)(21,23)(25,27)(26,28)(29,34)(30,35)(31,32)(33,36)(37,40)(38,39);(2,6)(4,7)(10,16)(11,18)(17,21)(19,22)(25,28,31,29)(26,32,34,27)(30,37,38,36)(33,35,40,39);(2,6)(3,8)(9,12,14,15)(10,11,16,18)(13,23,24,20)(17,22,21,19)(25,29,31,28)(26,32,34,27)(30,37,38,36)(33,39,40,35);(1,3)(2,4)(5,8)(6,7)(9,13)(10,17)(11,19)(12,20)(14,24)(15,23)(16,21)(18,22)(25,26)(27,29)(28,32)(30,33)(31,34)(35,37)(36,39)(38,40);(1,4)(2,3)(5,7)(6,8)(9,11)(10,15)(12,16)(13,19)(14,18)(17,23)(20,21)(22,24)(25,30,31,38)(26,33,34,40)(27,35,32,39)(28,36,29,37)Polycyclic presentation:
2^9|a^2=h,b^2=hi,c^2=g,d^2=gh,f^2=g,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dghi,e^b=egi,f^c=fgi,e^d=egi92 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8295134);