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 | #39467 | C2 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
4644 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)(4,6)(5,8)(9,10,16,19)(11,20,22,17)(12,15,13,23)(14,24,21,18)(25,26,30,34)(27,35,36,31)(28,32,37,39)(29,33,38,40);(1,2,4,6)(3,5,8,7)(9,11)(10,17)(12,14)(13,21)(15,18)(16,22)(19,20)(23,24)(27,36)(28,37)(31,35)(32,39);(1,2)(3,5)(4,6)(7,8)(9,12,16,13)(10,15,19,23)(11,14,22,21)(17,18,20,24)(25,27)(26,31)(28,33)(29,32)(30,36)(34,35)(37,40)(38,39);(1,3,4,8)(2,5,6,7)(9,13,16,12)(10,15,19,23)(11,21,22,14)(17,18,20,24)(25,28,30,37)(26,32,34,39)(27,33,36,40)(29,35,38,31);(9,14,16,21)(10,18,19,24)(11,12,22,13)(15,20,23,17);(9,15)(10,13)(11,18)(12,19)(14,20)(16,23)(17,21)(22,24)(25,29)(26,33)(27,32)(28,31)(30,38)(34,40)(35,37)(36,39)Polycyclic presentation:
2^9|a^2=gi,b^2=hi,c^2=ghi,d^2=gh,e^2=ghi,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, 8364154);