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 |
| prime radical | #45079 | C2 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
4341 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 40 points:
(2,5)(6,8)(25,26)(27,35)(28,33)(29,34)(30,31)(32,36)(37,40)(38,39);(1,2,4,5)(3,6,7,8)(9,10,14,15)(11,16,20,23)(12,13,17,22)(18,24,21,19)(27,36)(29,37)(32,35)(34,40);(10,15)(13,22)(16,23)(19,24)(25,27,31,36)(26,32,30,35)(28,37,39,29)(33,34,38,40);(9,11)(10,16)(12,21)(13,19)(14,20)(15,23)(17,18)(22,24)(25,28,31,39)(26,33,30,38)(27,29,36,37)(32,40,35,34);(9,12,14,17)(10,13,15,22)(11,18,20,21)(16,24,23,19)(25,29)(26,34)(27,28)(30,40)(31,37)(32,38)(33,35)(36,39);(1,3,4,7)(2,6,5,8)(9,13)(10,17)(11,19)(12,15)(14,22)(16,18)(20,24)(21,23)(25,30,31,26)(27,32,36,35)(28,38,39,33)(29,40,37,34)Polycyclic presentation:
2^9|b^2=ghi,c^2=h,d^2=h,e^2=hi,f^2=gh,b^a=bg,c^a=ch,f^a=fh,c^b=ci,e^b=eh,d^c=dh,f^c=fi,e^d=ei92 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7919182);