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 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
4724 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 40 points:
(1,2,4,6)(3,5,7,8)(9,10,14,15)(11,12,19,16)(13,17,22,23)(18,20,21,24)(25,26,30,35)(27,32,36,31)(28,33,39,38)(29,34,40,37);(3,7)(5,8)(10,15)(12,16)(17,23)(20,24)(27,36)(28,39)(31,32)(33,38);(1,2)(3,5)(4,6)(7,8)(25,27)(26,31)(28,37)(29,38)(30,36)(32,35)(33,40)(34,39);(1,3)(2,5)(4,7)(6,8)(9,11)(10,16)(12,15)(13,21)(14,19)(17,20)(18,22)(23,24)(25,27)(26,32)(28,37)(29,33)(30,36)(31,35)(34,39)(38,40);(1,2)(3,5)(4,6)(7,8)(9,12,14,16)(10,11,15,19)(13,20,22,24)(17,18,23,21)(25,28)(26,33)(27,37)(29,32)(30,39)(31,40)(34,36)(35,38);(9,13)(10,17)(11,18)(12,20)(14,22)(15,23)(16,24)(19,21)(25,29)(26,34)(27,38)(28,32)(30,40)(31,39)(33,36)(35,37)Polycyclic presentation:
2^9|a^2=gh,e^2=g,b^a=bg,c^a=ch,d^a=dg,e^a=eghi,c^b=ci,d^b=dh,e^b=egi,f^d=fgi83 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8378470);