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 | |
4736 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 40 points:
(2,5)(6,8)(9,10)(11,16)(12,20)(13,17)(14,19)(15,21)(18,22)(23,24)(25,26,31,34)(27,35,29,32)(28,33,39,30)(36,37,38,40);(2,5)(3,7)(12,21)(15,20)(18,24)(22,23)(25,26)(27,35)(28,30)(29,32)(31,34)(33,39)(36,40)(37,38);(1,2)(3,6)(4,5)(7,8)(9,11)(10,14)(12,18)(13,19)(15,22)(16,17)(20,23)(21,24)(25,27,31,29)(26,32,34,35)(28,36,39,38)(30,37,33,40);(1,3)(2,6)(4,7)(5,8)(9,12)(10,15)(11,18)(13,21)(14,22)(16,23)(17,20)(19,24)(25,28)(26,33)(27,36)(29,38)(30,34)(31,39)(32,40)(35,37);(9,11)(10,16)(12,18)(13,19)(14,17)(15,23)(20,22)(21,24)(25,29,31,27)(26,32,34,35)(28,38,39,36)(30,37,33,40);(9,10)(11,14)(12,15)(13,17)(16,19)(18,22)(20,21)(23,24)(25,30,31,33)(26,28,34,39)(27,37,29,40)(32,36,35,38)Polycyclic presentation:
2^9|a^2=g,c^2=g,e^2=g,f^2=g,b^a=bg,c^a=ch,d^a=dhi,c^b=ci,d^b=dh,f^e=fghi83 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8004978);