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 | #49972 | C2 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
4225 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 40 points:
(1,2,5,3)(4,7,8,6)(9,10)(11,13)(12,17)(14,16)(15,18)(19,22)(20,21)(23,24)(25,26,31,29)(27,32,37,38)(28,34,30,33)(35,40,36,39);(2,3)(6,7)(9,11,14,18)(10,15,16,13)(12,19,20,24)(17,23,21,22)(25,27)(26,32)(28,35)(29,38)(30,36)(31,37)(33,39)(34,40);(1,3,5,2)(4,7,8,6)(10,16)(12,20)(13,15)(19,24)(25,28,31,30)(26,33,29,34)(27,35,37,36)(32,39,38,40);(1,4,5,8)(2,6,3,7)(9,12,14,20)(10,17,16,21)(11,19,18,24)(13,22,15,23);(9,10)(11,15)(12,21)(13,18)(14,16)(17,20)(19,22)(23,24)(25,29,31,26)(27,32,37,38)(28,34,30,33)(35,39,36,40);(9,13,14,15)(10,18,16,11)(12,22,20,23)(17,24,21,19)(25,30,31,28)(26,34,29,33)(27,36,37,35)(32,40,38,39)Polycyclic presentation:
2^9|a^2=gh,b^2=gi,c^2=gh,d^2=g,e^2=ghi,f^2=h,b^a=bg,c^a=ch,d^a=di,f^a=fh,c^b=ci,e^b=eghi,d^c=dgi,e^c=egi,f^c=fgi,e^d=egi80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8961600);