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 | |
4494 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 40 points:
(9,10)(11,17)(12,18)(13,16)(14,15)(19,24)(20,21)(22,23)(25,26,30,33)(27,34,29,31)(28,32,37,39)(35,40,36,38);(1,2)(3,6)(4,5)(7,8)(10,15)(11,18)(16,23)(19,21)(25,27,30,29)(26,31,33,34)(28,35,37,36)(32,38,39,40);(2,5)(6,8)(10,15)(12,17)(16,23)(20,24)(25,28,30,37)(26,32,33,39)(27,35,29,36)(31,38,34,40);(3,7)(6,8)(9,11)(10,12)(13,21)(14,18)(15,17)(16,24)(19,22)(20,23)(25,26)(27,31)(28,32)(29,34)(30,33)(35,38)(36,40)(37,39);(1,2)(3,6)(4,5)(7,8)(9,12)(10,11)(13,20)(14,17)(15,18)(16,19)(21,23)(22,24)(25,28)(26,32)(27,35)(29,36)(30,37)(31,38)(33,39)(34,40);(1,3)(2,6)(4,7)(5,8)(9,13,14,22)(10,16,15,23)(11,19,18,21)(12,20,17,24)(25,29,30,27)(26,31,33,34)(28,36,37,35)(32,38,39,40)Polycyclic presentation:
2^9|a^2=gh,b^2=gh,c^2=gh,f^2=g,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,e^c=egi,f^d=fgi83 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8416700);