Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #486147 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C25 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C2×B(4,𝔽2) |
| commutator, Frattini, cosocle | C2×C22⋊C4 | C24 |
| prime radical | #45489 | C2 |
| LCS | G ▷ C2×C22⋊C4 ▷ C23 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C22×C4 ◁ C2×C4×D4 ◁ G | |
| LECS | G ▷ C2×C22⋊C4 ▷ C23 ▷ C2 ▷ C1 | |
905 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 40 points:
(1,2,5,8)(3,7,4,6)(9,10,14,17)(11,19,21,35)(12,16,25,33)(13,26,29,18)(15,27,31,22)(20,36,23,39)(24,40,28,34)(30,38,32,37);(2,6)(3,4)(7,8)(9,11)(10,15)(12,23)(13,27)(14,21)(16,32)(17,31)(18,35)(19,26)(20,25)(22,29)(24,37)(28,38)(30,33)(34,36)(39,40);(9,12,14,25)(10,16,17,33)(11,20,21,23)(13,28,29,24)(15,30,31,32)(18,34,26,40)(19,36,35,39)(22,38,27,37);(1,3,5,4)(2,6,8,7)(10,17)(11,21)(16,33)(18,26)(20,23)(22,27)(34,40)(37,38)Polycyclic presentation:
2^9|a^2=g,c^2=i,d^2=gi,e^2=g,b^a=be,c^a=cf,d^a=dg,e^a=eg,h^a=hi,c^b=ci,d^b=dg,e^b=eg,f^b=fhi,e^c=eh,f^e=fi47 conjugacy classes — view the full character table.
| maximal subgroup of | #72280914, #72281012, #72281090, … |
| maximal normal subgroup of | #72280914, #72281012, #72281090, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 393554);