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 | |
4375 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)(3,7)(4,5)(6,8)(9,10)(11,16)(12,17)(13,18)(14,15)(19,23)(20,24)(21,22)(25,26,31,29)(27,35,37,32)(28,38,39,33)(30,40,36,34);(2,5)(6,7)(10,15)(16,17)(18,21)(23,24)(25,27,31,37)(26,32,29,35)(28,36,39,30)(33,34,38,40);(1,2)(3,6)(4,5)(7,8)(9,11,14,12)(10,16,15,17)(13,19,22,20)(18,23,21,24)(25,28,31,39)(26,33,29,38)(27,30,37,36)(32,40,35,34);(1,3,4,8)(2,6,5,7)(9,12,14,11)(10,17,15,16)(13,20,22,19)(18,24,21,23)(25,28)(26,33)(27,36)(29,38)(30,37)(31,39)(32,34)(35,40);(9,10)(11,16)(12,17)(13,21)(14,15)(18,22)(19,24)(20,23)(25,29,31,26)(27,32,37,35)(28,38,39,33)(30,40,36,34);(9,13)(10,18)(11,19)(12,20)(14,22)(15,21)(16,23)(17,24)(25,30,31,36)(26,34,29,40)(27,28,37,39)(32,38,35,33)Polycyclic presentation:
2^9|a^2=h,b^2=h,c^2=ghi,d^2=gh,e^2=h,f^2=h,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 | #72219281, #72219283 |
| maximal quotient of | #72219273 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 9825275);