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 | |
4415 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)(11,15)(12,17)(13,18)(14,19)(16,21)(20,23)(22,24)(25,26,30,33)(27,31,36,39)(28,29,37,32)(34,35,40,38);(2,5)(6,7)(9,11)(10,15)(12,18)(13,17)(14,21)(16,19)(20,22)(23,24)(25,27,30,36)(26,31,33,39)(28,34,37,40)(29,35,32,38);(1,2)(3,6)(4,5)(7,8)(9,11,14,21)(10,15,19,16)(12,20,22,18)(13,17,23,24)(25,28,30,37)(26,32,33,29)(27,34,36,40)(31,38,39,35);(1,3,4,8)(2,6,5,7)(9,11)(10,16)(12,20)(13,17)(14,21)(15,19)(18,22)(23,24)(25,28)(26,29)(27,34)(30,37)(31,35)(32,33)(36,40)(38,39);(9,12)(10,17)(11,20)(13,16)(14,22)(15,23)(18,21)(19,24)(25,27)(26,31)(28,34)(29,35)(30,36)(32,38)(33,39)(37,40);(9,13)(10,18)(11,17)(12,15)(14,23)(16,22)(19,20)(21,24)(25,29)(26,28)(27,35)(30,32)(31,34)(33,37)(36,38)(39,40)Polycyclic presentation:
2^9|a^2=h,b^2=h,c^2=ghi,d^2=g,b^a=bg,c^a=ch,d^a=di,f^a=fh,c^b=ci,e^b=eghi,d^c=dghi,f^e=fgi77 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 9877284);