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 | #392 | C23 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
4111 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 40 points:
(9,10)(11,19)(12,13)(14,15)(16,22)(17,18)(20,21)(23,24)(25,26,31,35)(27,29,33,40)(28,34,30,32)(36,39,37,38);(1,2,5,3)(4,7,8,6)(10,15)(13,17)(16,19)(21,24)(25,27)(26,29)(28,36)(30,37)(31,33)(32,38)(34,39)(35,40);(2,3)(6,7)(9,11,14,22)(10,16,15,19)(12,23,18,20)(13,21,17,24)(25,28,31,30)(26,32,35,34)(27,36,33,37)(29,38,40,39);(1,2,5,3)(4,6,8,7)(9,12,14,18)(10,17,15,13)(11,20,22,23)(16,21,19,24);(1,3,5,2)(4,7,8,6)(9,10)(11,19)(12,17)(13,18)(14,15)(16,22)(20,24)(21,23)(25,29)(26,33)(27,35)(28,38)(30,39)(31,40)(32,37)(34,36);(1,4,5,8)(2,6,3,7)(9,13,14,17)(10,18,15,12)(11,21,22,24)(16,20,19,23)(25,30,31,28)(26,34,35,32)(27,37,33,36)(29,39,40,38)Polycyclic presentation:
2^9|a^2=gh,b^2=i,c^2=h,d^2=gi,e^2=i,f^2=hi,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,f^b=fgi,d^c=dgi,e^c=egi86 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8731072);