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 | |
4488 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 40 points:
(2,4)(6,8)(9,10)(11,18)(12,20)(13,17)(14,15)(16,21)(19,23)(22,24);(3,7)(6,8)(9,11,15,17)(10,13,14,18)(12,19,21,24)(16,23,20,22)(25,26,30,31)(27,32,35,40)(28,33,38,37)(29,34,39,36);(1,2,5,4)(3,6,7,8)(10,14)(12,21)(13,18)(19,24)(26,31)(27,35)(28,38)(29,39);(9,12,15,21)(10,16,14,20)(11,19,17,24)(13,23,18,22)(25,27)(26,32)(28,34)(29,37)(30,35)(31,40)(33,39)(36,38);(1,3,5,7)(2,6,4,8)(9,13,15,18)(10,17,14,11)(12,22,21,23)(16,19,20,24)(25,28)(26,33)(27,36)(29,32)(30,38)(31,37)(34,35)(39,40);(1,4,5,2)(3,8,7,6)(9,14,15,10)(11,18,17,13)(12,20,21,16)(19,22,24,23)(25,29)(26,34)(27,37)(28,32)(30,39)(31,36)(33,35)(38,40)Polycyclic presentation:
2^9|b^2=gi,c^2=gh,d^2=g,e^2=h,f^2=h,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,d^c=dgi,e^c=egi,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, 8886494);