#106621

order	512
nilpotency class	3
isoclinism	isoclinic class of #8346
automorphism	?
homology	Abelianization (H₁) = C₂×C₄² · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	#379
commutator	C₂²×C₄	C₂×C₄²
Frattini, cosocle	C₂²×C₄²	C₂³
socle	C₂²	C₂²⋊#95
prime radical	C₂²×#41	D₄
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂²×C₄² ▷ C₂² ▷ C₁

916 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:

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

Polycyclic presentation:

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

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, 13817);