Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #486155 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C26 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C2×B(4,𝔽2) |
| commutator, Frattini, cosocle | C2×C4⋊C4 | C24 |
| LCS | G ▷ C2×C4⋊C4 ▷ C23 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C22×C4 ◁ C2×C4×D4 ◁ G | |
| LECS | G ▷ C2×C4⋊C4 ▷ C23 ▷ C2 ▷ C1 | |
998 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)(3,7)(4,6)(5,8)(9,10)(11,15)(12,22)(13,25)(14,18)(16,24)(17,27)(19,34)(20,32)(21,30)(23,33)(26,31)(28,36)(29,38)(35,39)(37,40);(2,6)(3,4)(7,8)(9,11,14,21)(10,15,18,30)(12,19,23,35)(13,20,27,38)(16,28,31,40)(17,32,25,29)(22,39,33,34)(24,36,26,37);(9,12)(10,16)(11,19)(13,26)(14,23)(15,28)(17,33)(18,31)(20,37)(21,35)(22,25)(24,27)(29,39)(30,40)(32,34)(36,38);(1,3,5,4)(2,6,8,7)(12,23)(16,31)(19,35)(22,33)(24,26)(28,40)(34,39)(36,37)Polycyclic presentation:
2^9|b^2=i,d^2=g,e^2=g,f^2=i,b^a=be,c^a=cf,d^a=dg,e^a=eg,f^a=fi,h^a=hi,d^b=dg,e^b=eg,f^b=fhi,d^c=di,e^c=eh,f^c=fi,f^e=fi47 conjugacy classes — view the full character table.
| maximal subgroup of | #72240242, #72240247, #72280943, … |
| maximal normal subgroup of | #72240242, #72240247, #72280943, … |
| maximal quotient of | #72240330 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 393480);