Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #486091 |
| 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 |
| 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 | |
981 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)(11,19)(12,16)(13,22)(14,17)(15,20)(18,23)(21,24);(1,2)(3,6)(4,7)(5,8)(9,11,14,12)(10,15,17,24)(13,21,23,20)(16,22,19,18);(1,3,5,4)(2,6,8,7)(9,12,14,11)(10,16,17,19)(13,20,23,21)(15,22,24,18);(1,3,5,4)(2,7,8,6)(10,17)(11,12)(18,22)(20,21)Polycyclic presentation:
2^9|b^2=i,c^2=g,d^2=gi,e^2=g,b^a=be,c^a=cf,d^a=dg,e^a=eg,h^a=hi,d^b=dg,e^b=eg,f^b=fhi,d^c=di,e^c=eh,f^e=fi47 conjugacy classes — view the full character table.
| maximal subgroup of | #72240202, #72240204 |
| maximal normal subgroup of | #72240202, #72240204 |
| maximal quotient of | #72240328 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 393716);