Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #419810 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C23×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #2668 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| prime radical | C24 | H(5,𝔽2) |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
862 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 4 generators on 40 points:
(1,2,4,6)(3,7,8,5)(9,10)(11,22)(12,27)(13,20)(14,19)(15,21)(16,24)(17,25)(18,30)(23,32)(26,33)(28,38)(29,37)(31,36)(34,40)(35,39);(1,3,4,8)(2,5,6,7)(9,11)(10,16)(12,28)(13,25)(14,24)(15,26)(17,21)(18,35)(19,33)(20,22)(23,29)(27,34)(30,40)(31,39)(32,36)(37,38);(9,11)(10,17,21,16)(12,23)(13,24)(14,25)(15,26)(18,34,37,32)(19,33,20,22)(27,35,36,38)(28,29)(30,39)(31,40);(3,8)(5,7)(9,12,15,31)(10,18,21,37)(11,23,26,40)(13,30,14,29)(16,32,17,34)(19,27,20,36)(22,38,33,35)(24,39,25,28)Polycyclic presentation:
2^9|a^2=ef,b^2=ef,c^2=g,d^2=i,e^2=i,f^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,f^a=fi,g^a=gi,h^a=hi,c^b=cg,d^b=de,e^b=ei,f^b=fi,h^c=hi,e^d=ei,f^d=fi,h^d=hi62 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 351201);