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 | #59803 | C2 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
4715 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 40 points:
(1,2,5,3)(4,7,8,6)(12,20)(16,23)(18,24)(21,22)(26,31)(27,33)(28,36)(35,40);(2,3)(6,7)(9,10)(11,15)(12,16)(13,17)(14,19)(18,22)(20,23)(21,24)(25,26,30,31)(27,32,33,38)(28,37,36,29)(34,40,39,35);(1,3,5,2)(4,7,8,6)(9,11)(10,14)(12,18)(13,19)(15,17)(16,21)(20,24)(22,23)(25,27)(26,32)(28,34)(29,35)(30,33)(31,38)(36,39)(37,40);(1,4)(2,6)(3,7)(5,8)(9,11)(10,15)(12,18)(13,19)(14,17)(16,22)(20,24)(21,23)(25,28,30,36)(26,29,31,37)(27,34,33,39)(32,35,38,40);(9,12)(10,16)(11,18)(13,20)(14,21)(15,22)(17,23)(19,24);(9,10)(11,14)(12,16)(13,17)(15,19)(18,21)(20,23)(22,24)(25,29)(26,28)(27,35)(30,37)(31,36)(32,34)(33,40)(38,39)Polycyclic presentation:
2^9|a^2=gh,b^2=h,c^2=gh,d^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, 8337738);