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 | #45978 | C2 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
4256 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 40 points:
(3,7)(6,8)(9,10,14,18)(11,17,13,15)(12,16,20,23)(19,24,21,22)(25,26,30,31)(27,28,35,32)(29,33,37,38)(34,36,40,39);(1,2)(3,6)(4,5)(7,8)(9,11,14,13)(10,15,18,17)(12,19,20,21)(16,22,23,24)(26,31)(28,32)(33,38)(36,39);(2,5)(6,8)(12,20)(16,23)(19,21)(22,24)(25,27)(26,32)(28,31)(29,34)(30,35)(33,39)(36,38)(37,40);(9,10)(11,15)(12,16)(13,17)(14,18)(19,22)(20,23)(21,24)(25,28,30,32)(26,27,31,35)(29,36,37,39)(33,34,38,40);(1,2)(3,6)(4,5)(7,8)(9,12)(10,16)(11,19)(13,21)(14,20)(15,22)(17,24)(18,23);(1,3)(2,6)(4,7)(5,8)(9,13,14,11)(10,17,18,15)(12,21,20,19)(16,24,23,22)(25,29,30,37)(26,33,31,38)(27,34,35,40)(28,36,32,39)Polycyclic presentation:
2^9|a^2=g,b^2=gh,d^2=h,f^2=g,b^a=bg,c^a=ch,d^a=dg,f^a=fghi,c^b=ci,d^b=dh,e^c=eghi86 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8000907);