Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #486155 |
| 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×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 | |
916 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 24 points:
(2,6)(3,4)(7,8)(9,10)(11,18)(12,15)(13,16)(14,19)(17,21)(20,24)(22,23);(1,2)(3,6)(4,7)(5,8)(10,15)(12,21)(16,23)(17,19)(18,24)(20,22);(1,3,5,4)(2,6,8,7)(10,16)(11,13)(12,21)(15,23)(17,24)(18,19);(1,3,5,4)(2,7,8,6)(9,11,14,13)(10,16,19,18)(12,20,21,22)(15,23,17,24)Polycyclic presentation:
2^9|c^2=gi,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=dgi,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 | #72282376, #72282385, #72282424, … |
| maximal normal subgroup of | #72282376, #72282385, #72282424, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 393761);