Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of D4×B(4,𝔽2) |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C29 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C22×#120 |
| commutator, Frattini, cosocle | C24 | C25 |
| LCS | G ▷ C24 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C22×C22⋊C4 ◁ G | |
| LECS | G ▷ C24 ▷ C2 ▷ C1 | |
3414 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 5 generators on 24 points:
(9,10)(11,17,19,14)(12,15)(13,16)(18,23,24,21)(20,22);(1,2,4,5)(3,6,7,8)(9,10)(11,14)(12,15)(13,16)(17,19)(18,21)(20,22)(23,24);(2,5)(3,7)(9,11)(10,14)(12,18)(13,19)(15,21)(16,17)(20,24)(22,23);(1,3)(2,6)(4,7)(5,8)(9,12,13,20)(10,15,16,22)(11,18,19,24)(14,21,17,23);(2,5)(3,7)Polycyclic presentation:
2^9|a^2=f,b^2=h,d^2=i,b^a=bf,c^a=cg,g^a=gi,c^b=ch,e^b=eh,g^b=gi,d^c=dh,f^c=fi,e^d=eh80 conjugacy classes — view the full character table.
| maximal subgroup of | #72268316, #72268399, #72268427, … |
| maximal normal subgroup of | #72268316, #72268399, #72268427, … |
| maximal quotient of | #72311708 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6503382);