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 | |
4221 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 40 points:
(1,2,5,3)(4,6,8,7)(9,10)(11,14)(12,15)(13,21)(16,18)(17,24)(19,20)(22,23)(25,26)(27,31)(28,37)(29,33)(30,34)(32,36)(35,40)(38,39);(2,3)(6,7)(9,11,16,15)(10,14,18,12)(13,19,24,23)(17,22,21,20)(27,35)(29,38)(31,40)(33,39);(9,12,16,14)(10,11,18,15)(13,22,24,20)(17,23,21,19)(25,27,30,35)(26,31,34,40)(28,29,36,38)(32,39,37,33);(9,13,16,24)(10,17,18,21)(11,19,15,23)(12,20,14,22)(25,28,30,36)(26,32,34,37)(27,29,35,38)(31,39,40,33);(1,3,5,2)(4,6,8,7)(9,14,16,12)(10,11,18,15)(13,22,24,20)(17,19,21,23)(25,29)(26,33)(27,36)(28,35)(30,38)(31,32)(34,39)(37,40);(1,4,5,8)(2,6,3,7)(9,15,16,11)(10,14,18,12)(13,23,24,19)(17,22,21,20)(25,26)(27,31)(28,32)(29,39)(30,34)(33,38)(35,40)(36,37)Polycyclic presentation:
2^9|a^2=g,b^2=h,c^2=i,d^2=i,e^2=gh,f^2=gh,b^a=bg,c^a=ch,d^a=di,f^a=fh,c^b=ci,e^b=eghi,d^c=dh,f^e=fgi77 conjugacy classes — view the full character table.
| maximal normal subgroup of | #72219279, #72219283 |
| maximal quotient of | #72219276 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 9831102);