#105689

order	512
nilpotency class	3
isoclinism	isoclinic class of #105688
automorphism	?
homology	Abelianization (H₁) = C₂×C₄² · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	C₄×C₂²⋊C₄
commutator	C₄²	C₂×C₄²
Frattini, cosocle	C₂²×C₄²	C₂³
socle	C₂²	C₂²⋊#95
prime radical	C₂⁴	#120
LCS	G ▷ C₄² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₄×#117 ◁ G
LECS	G ▷ C₂²×C₄² ▷ C₂² ▷ C₁

929 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)(3,7)(4,8)(9,10)(11,17,19,23)(12,15)(13,16)(14,18,21,24)(20,22);(1,2,5,3)(4,7,8,6)(9,11)(10,14,16,21)(12,18)(13,19)(15,17,22,23)(20,24);(1,3,5,2)(4,6,8,7)

Polycyclic presentation:

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

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 12885);