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 | #40720 | C2 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
4608 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)(10,15)(11,17)(12,19)(20,24)(25,26,30,34)(27,32,35,31)(28,29,33,39)(36,38,40,37);(2,5)(6,7)(9,10,14,15)(11,13,17,21)(12,16,19,22)(18,20,23,24)(27,35)(31,32)(36,40)(37,38);(1,2)(3,6)(4,5)(7,8)(25,27)(26,31)(28,36)(29,37)(30,35)(32,34)(33,40)(38,39);(1,3)(2,6)(4,8)(5,7)(9,11)(10,13)(12,18)(14,17)(15,21)(16,20)(19,23)(22,24)(25,27)(26,32)(28,36)(29,38)(30,35)(31,34)(33,40)(37,39);(9,12)(10,16)(11,18)(13,20)(14,19)(15,22)(17,23)(21,24)(25,28,30,33)(26,29,34,39)(27,36,35,40)(31,37,32,38);(9,13)(10,17)(11,15)(12,20)(14,21)(16,23)(18,22)(19,24)(25,29)(26,33)(27,37)(28,34)(30,39)(31,40)(32,36)(35,38)Polycyclic presentation:
2^9|a^2=h,b^2=ghi,e^2=h,b^a=bg,c^a=ch,d^a=dg,e^a=eghi,c^b=ci,d^b=dh,f^d=fgi89 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8341916);