#109586

order	512
nilpotency class	3
isoclinism	isoclinic class of B(4,𝔽₂)
automorphism	?
homology	Abelianization (H₁) = C₂×C₄×C₈ · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄²	#120
commutator	C₂³	C₂×C₄×C₈
Frattini, cosocle	C₂²×C₄²	C₂³
socle	C₂²	#1436
prime radical	C₂⁴	C₂×C₄²
LCS	G ▷ C₂³ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂²×C₄² ▷ C₂² ▷ C₁

618 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 24
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 3 generators on 24 points:

(1,2,5,3,4,6,8,7)(14,20)(21,24);(9,10,12,16,13,15,19,22)(11,14,18,21,17,20,23,24);(1,3,5,6,4,7,8,2)(9,11)(10,14)(12,18)(13,17)(15,20)(16,21)(19,23)(22,24)

Polycyclic presentation:

2^9|a^2=f,b^2=g,c^2=f,f^2=i,g^2=h,b^a=bd,c^a=ce,e^b=eh,d^c=dh

Conjugacy classes & character table

128 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 16782);