Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #62307 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C4 · Schur multiplier (H2) = C28 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | C23×D4 |
| commutator | C2×C4 | C24×C4 |
| Frattini, cosocle | C22×C4 | C25 |
| socle | C22 | C22×C4×D4 |
| prime radical | #61124 | C2 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C2×#435 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1770 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 5 generators on 40 points:
(1,2)(3,7)(4,6)(5,8)(9,10,16,20,14,19,15,32)(11,21,26,36,23,17,25,35)(12,27,13,30,28,40,29,18)(22,37,31,33,38,39,24,34);(1,3)(2,5)(4,8)(6,7)(9,11)(10,17)(12,22)(13,31)(14,23)(15,25)(16,26)(18,33)(19,21)(20,35)(24,29)(27,39)(28,38)(30,34)(32,36)(37,40);(2,6)(5,7)(9,12,16,13,14,28,15,29)(10,18,32,40,19,30,20,27)(11,22,26,31,23,38,25,24)(17,33,36,37,21,34,35,39);(3,8)(5,7)(11,23)(17,21)(22,38)(24,31)(25,26)(33,34)(35,36)(37,39);(2,6)(5,7)(9,13,16,28,14,29,15,12)(10,18,20,27,19,30,32,40)(11,24,26,22,23,31,25,38)(17,34,35,37,21,33,36,39)Polycyclic presentation:
2^9|a^2=h,c^2=fg,e^2=h,g^2=i,h^2=i,b^a=bf,c^a=cg,e^a=ef,g^a=gi,d^b=df,e^b=ei,e^c=ei98 conjugacy classes — view the full character table.
| maximal subgroup of | #72265375, #72308712, #72308766 |
| maximal normal subgroup of | #72265375, #72308712, #72308766 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6310923);