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 | #39278 | C2 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
4371 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 40 points:
(2,3)(6,7)(9,10,15,20)(11,14,19,16)(12,13,18,17)(21,23,24,22)(25,26,30,34)(27,35,36,31)(28,32,37,39)(29,33,38,40);(1,2,5,3)(4,6,8,7)(9,11)(10,16)(12,21)(13,22)(14,20)(15,19)(17,23)(18,24)(27,36)(28,37)(31,35)(32,39);(9,12,15,18)(10,17,20,13)(11,21,19,24)(14,22,16,23)(25,27)(26,31)(28,33)(29,32)(30,36)(34,35)(37,40)(38,39);(1,2,5,3)(4,7,8,6)(9,10)(11,14)(12,17)(13,18)(15,20)(16,19)(21,22)(23,24)(25,28,30,37)(26,32,34,39)(27,33,36,40)(29,35,38,31);(1,3,5,2)(4,7,8,6)(9,13,15,17)(10,18,20,12)(11,22,19,23)(14,21,16,24);(1,4)(2,6)(3,7)(5,8)(9,14)(10,19)(11,20)(12,22)(13,21)(15,16)(17,24)(18,23)(25,29)(26,33)(27,32)(28,31)(30,38)(34,40)(35,37)(36,39)Polycyclic presentation:
2^9|a^2=h,b^2=ghi,c^2=hi,d^2=gh,e^2=g,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, 8343815);