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 | |
4740 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)(3,7)(4,5)(6,8)(9,10,14,12)(11,17,18,15)(13,20,21,16)(19,22,24,23)(25,26)(27,32)(28,31)(29,33)(30,34)(35,39)(36,38)(37,40);(2,5)(6,7)(10,12)(15,17)(16,20)(22,23)(25,27,30,28)(26,31,34,32)(29,35,37,36)(33,38,40,39);(1,2)(3,6)(4,5)(7,8)(10,12)(11,18)(13,21)(22,23)(25,27,30,28)(26,32,34,31)(29,36,37,35)(33,38,40,39);(1,3,4,8)(2,6,5,7)(9,11)(10,15)(12,17)(13,19)(14,18)(16,22)(20,23)(21,24)(25,28,30,27)(26,32,34,31)(29,35,37,36)(33,38,40,39);(9,12,14,10)(11,15,18,17)(13,20,21,16)(19,22,24,23)(25,29)(26,33)(27,35)(28,36)(30,37)(31,38)(32,39)(34,40);(9,13,14,21)(10,16,12,20)(11,19,18,24)(15,22,17,23)(25,26)(27,31)(28,32)(29,33)(30,34)(35,38)(36,39)(37,40)Polycyclic presentation:
2^9|a^2=h,b^2=ghi,c^2=ghi,d^2=gh,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=dh,e^c=egi,f^c=fgi,e^d=egi80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 9405604);