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) = C210 |
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 ◁ C23×D4 ◁ G | |
| LECS | G ▷ C24 ▷ C2 ▷ C1 | |
5101 conjugacy classes of subgroups.
| minimal degree | μ(G) = 16 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 5 generators on 16 points:
(13,16);(1,2,4,5)(3,6,7,8)(9,10)(11,13)(12,14)(15,16);(2,5)(3,7)(9,11)(10,13)(12,15)(14,16);(1,3)(2,6)(4,7)(5,8);(2,5)(3,7)Polycyclic presentation:
2^9|b^2=h,b^a=bf,c^a=cg,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 | #72216616, #72220624, #72234132, … |
| maximal normal subgroup of | #72216616, #72234132, #72234532, … |
| maximal quotient of | #72235081, #72236627, C2×#6595772, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6502968);