Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #454676 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C23×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #2684 |
| 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 | |
812 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 40 points:
(1,2,5,7)(3,8,4,6)(9,10,13,17,15,20,14,19)(11,21,24,32,22,31,25,16)(12,26,28,34,29,33,27,18)(23,37,38,40,36,30,39,35);(9,11,15,22)(10,16,20,32)(12,23,29,36)(13,25,14,24)(17,31,19,21)(18,30,34,37)(26,40,33,35)(27,38,28,39);(1,3,5,4)(2,6,7,8)(10,17)(11,22)(12,27)(13,14)(16,21)(19,20)(23,38)(26,33)(28,29)(31,32)(35,40)(36,39);(1,4,5,3)(2,6,7,8)(9,12)(10,18)(11,23)(13,28)(14,27)(15,29)(16,30)(17,26)(19,33)(20,34)(21,35)(22,36)(24,38)(25,39)(31,40)(32,37)Polycyclic presentation:
2^9|a^2=fg,b^2=i,c^2=ef,d^2=ef,e^2=i,f^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,h^a=hi,c^b=cg,e^b=ei,f^b=fi,h^b=hi,d^c=de,e^c=ei,f^c=fi,h^c=hi,e^d=ei,f^d=fi,g^d=gi,h^d=hi56 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 370854);