#109351

order	512
nilpotency class	3
isoclinism	isoclinic class of #1685
automorphism	?
homology	Abelianization (H₁) = C₂×C₄² · Schur multiplier (H₂) = C₂²×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄²	#121
commutator	C₄²	C₂×C₄²
Frattini, cosocle	C₂²×C₄²	C₂³
socle	C₂²	C₄⋊#95
prime radical	C₂⁴	C₄⋊Q₈
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₄² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂²×C₄² ▷ C₂² ▷ C₁

583 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:

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

Polycyclic presentation:

2^9|a^2=f,b^2=g,c^2=e,d^2=hi,e^2=h,f^2=h,g^2=hi,b^a=bd,c^a=ce,d^a=dh,e^a=eh,g^a=gh,c^b=ch,d^b=di,f^b=fi

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 16547);