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 | |
4574 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)(9,10,14,17)(11,15,20,23)(12,13,21,16)(18,19,24,22)(25,26)(27,29)(28,35)(30,33)(31,38)(32,39)(34,40)(36,37);(1,3,5,8)(2,6,4,7)(9,11,14,20)(10,15,17,23)(12,18,21,24)(13,19,16,22)(25,27,30,32)(26,29,33,39)(28,36,38,34)(31,40,35,37);(9,12,14,21)(10,16,17,13)(11,18,20,24)(15,22,23,19)(25,28,30,38)(26,31,33,35)(27,34,32,36)(29,37,39,40);(1,2,5,4)(3,7,8,6)(9,12)(10,13)(11,18)(14,21)(15,19)(16,17)(20,24)(22,23)(25,28)(26,31)(27,34)(29,37)(30,38)(32,36)(33,35)(39,40);(1,3)(2,7)(4,6)(5,8)(9,11)(10,15)(12,18)(13,19)(14,20)(16,22)(17,23)(21,24)(25,26)(27,29)(28,31)(30,33)(32,39)(34,37)(35,38)(36,40);(1,4,5,2)(3,7,8,6)(9,13)(10,12)(11,19)(14,16)(15,18)(17,21)(20,22)(23,24)(25,29)(26,32)(27,33)(28,37)(30,39)(31,36)(34,35)(38,40)Polycyclic presentation:
2^9|a^2=ghi,b^2=gh,c^2=h,d^2=g,f^2=g,b^a=bg,c^a=ch,d^a=di,f^a=fh,c^b=ci,d^b=dgi,e^b=eghi,d^c=dh,f^e=fgi83 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 9843518);