#106759

order	512
nilpotency class	3
isoclinism	isoclinic class of #106738
automorphism	?
homology	Abelianization (H₁) = C₂×C₄² · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

No aliases for this group.

Subgroups

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

862 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 64

A minimal faithful permutation representation — 3 generators on 32 points:

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

Polycyclic presentation:

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

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 13955);