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 | |
4635 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 40 points:
(1,2,5,4)(3,7,8,6)(9,10)(11,14)(12,19)(13,15)(16,20)(17,23)(18,21)(22,24)(25,26)(27,32)(28,38)(29,34)(30,31)(33,39)(35,40)(36,37);(1,3,5,8)(2,6,4,7)(9,11,13,18)(10,14,15,21)(12,17,20,24)(16,22,19,23)(25,27)(26,32)(28,29)(30,36)(31,37)(33,40)(34,38)(35,39);(10,15)(14,21)(16,19)(22,23)(25,28,31,39)(26,33,30,38)(27,35,37,29)(32,34,36,40);(9,10)(11,14)(12,16)(13,15)(17,22)(18,21)(19,20)(23,24)(25,28)(26,33)(27,29)(30,38)(31,39)(32,40)(34,36)(35,37);(1,3)(2,7)(4,6)(5,8)(9,11)(10,14)(12,17)(13,18)(15,21)(16,22)(19,23)(20,24)(25,29)(26,34)(27,28)(30,40)(31,35)(32,38)(33,36)(37,39);(1,4,5,2)(3,7,8,6)(9,12,13,20)(10,16,15,19)(11,17,18,24)(14,22,21,23)(25,30,31,26)(27,36,37,32)(28,38,39,33)(29,34,35,40)Polycyclic presentation:
2^9|a^2=g,b^2=ghi,c^2=i,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, #72219281, #72219285 |
| maximal quotient of | #72219270 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 9827538);