Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #417682 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C24 |
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 | |
851 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 4 generators on 40 points:
(1,2,5,4)(3,7,8,6)(10,16)(11,21)(12,26)(13,14)(17,22)(18,34)(19,20)(23,37)(24,25)(28,29)(32,40)(36,38);(1,3,5,8)(2,6,4,7)(9,10)(11,22)(12,27)(13,19)(14,16)(15,20)(17,25)(18,26)(21,30)(23,38)(24,33)(28,34)(29,31)(32,39)(35,36)(37,40);(9,11,15,25)(10,17,20,22)(12,23,29,37)(13,21,14,24)(16,30,19,33)(18,32,34,36)(26,39,28,35)(27,38,31,40);(1,4,5,2)(3,6,8,7)(9,12)(10,18)(11,23)(13,28)(14,26)(15,29)(16,31)(17,32)(19,27)(20,34)(21,35)(22,36)(24,39)(25,37)(30,40)(33,38)Polycyclic presentation:
2^9|a^2=ef,b^2=ef,c^2=i,d^2=ef,e^2=i,f^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,f^a=fi,h^a=hi,c^b=cg,d^b=de,e^b=ei,f^b=fi,e^c=ei,f^c=fi,h^c=hi,e^d=ei,f^d=fi,g^d=gi,h^d=hi62 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 337745);