#382339

order	512
nilpotency class	2
isoclinism	isoclinic class of #16249
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₄²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, Frattini, cosocle	C₂³×C₄	C₂⁴
commutator, socle, prime radical	C₂⁴	C₂³×C₄
LCS	G ▷ C₂⁴ ▷ C₁
UCS	C₁ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

876 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 512

A minimal faithful permutation representation — 4 generators on 32 points:

(1,2)(3,6)(4,5)(7,8)(9,10)(11,14)(12,16)(13,15)(25,26,29,30,28,31,27,32);(2,5)(6,8)(9,11,13,12)(10,14,15,16)(17,18,20,22)(19,23,24,21)(25,27,28,29)(26,30,31,32);(1,3,4,7)(2,6,5,8)(10,15)(14,16)(17,19,20,24)(18,21,22,23)(25,27,28,29)(26,30,31,32);(3,7)(6,8)(9,12,13,11)(10,16,15,14)(25,26,27,32,28,31,29,30)

Polycyclic presentation:

2^9|a^2=h,b^2=fg,c^2=eg,d^2=fh,h^2=i,b^a=be,c^a=cf,d^a=di,c^b=cg,d^b=di,d^c=de

Conjugacy classes & character table

104 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 289535);