Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #476112 |
| 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 | |
743 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 4 generators on 40 points:
(1,2,5,3)(4,7,8,6)(9,10)(11,21)(12,25)(13,20)(14,18)(15,19)(16,23)(17,28)(22,32)(24,36)(26,37)(27,35)(29,34)(30,31)(33,39)(38,40);(1,3,5,2)(4,7,8,6)(9,11)(10,16)(12,26)(13,30)(14,23)(15,24)(17,33)(18,36)(19,31)(20,21)(22,27)(25,40)(28,38)(29,39)(32,34)(35,37);(1,4,5,8)(2,6,3,7)(12,27)(13,14)(16,31)(17,34)(18,20)(22,26)(23,30)(25,35)(28,29)(32,37)(33,40)(38,39);(2,3)(6,7)(9,12,15,29)(10,17,19,35)(11,22,24,38)(13,28,14,27)(16,32,31,40)(18,25,20,34)(21,37,36,33)(23,39,30,26)Polycyclic presentation:
2^9|a^2=f,b^2=f,c^2=f,d^2=i,e^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,g^a=gi,h^a=hi,c^b=cg,d^b=def,e^b=ei,d^c=de,e^c=ei,h^c=hi,e^d=ei,h^d=hi50 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 384952);