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 | |
4614 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 40 points:
(1,2,5,6)(3,4,7,8)(9,10)(11,17)(12,19)(13,21)(14,15)(16,23)(18,20)(22,24)(25,26)(27,28)(29,32)(30,33)(31,35)(34,36)(37,38)(39,40);(2,6)(3,7)(10,15)(12,17)(16,21)(20,24)(27,31)(28,35)(34,39)(36,40);(1,3)(2,4)(5,7)(6,8)(9,11)(10,12)(13,18)(14,19)(15,17)(16,20)(21,24)(22,23)(25,27,30,31)(26,28,33,35)(29,34,38,39)(32,36,37,40);(1,4)(2,3)(5,8)(6,7)(9,11)(10,12)(13,22)(14,19)(15,17)(16,24)(18,23)(20,21)(25,26)(27,28)(29,37)(30,33)(31,35)(32,38)(34,40)(36,39);(9,12)(10,11)(13,20)(14,17)(15,19)(16,18)(21,22)(23,24)(25,28)(26,31)(27,33)(29,36)(30,35)(32,39)(34,37)(38,40);(9,13,14,23)(10,16,15,21)(11,18,19,22)(12,20,17,24)(25,29)(26,32)(27,34)(28,36)(30,38)(31,39)(33,37)(35,40)Polycyclic presentation:
2^9|a^2=gh,c^2=ghi,f^2=h,b^a=bg,c^a=ch,d^a=dg,e^a=egi,f^a=fh,c^b=ci,e^b=egi,e^d=eghi,f^d=fgi83 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 10331866);