#106853

order	512
nilpotency class	3
isoclinism	isoclinic class of #9345
automorphism	?
homology	Abelianization (H₁) = C₂×C₄² · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	#386
commutator	C₂²×C₄	C₂×C₄²
Frattini, cosocle	C₂⁴×C₄	C₂³
socle	C₂²	C₂²⋊#95
prime radical	C₂⁵	#41
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂⁴×C₄ ◁ G
LECS	G ▷ C₂⁴×C₄ ▷ C₂² ▷ C₁

918 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)(3,9)(4,7)(5,8)(6,11)(10,15)(12,14)(13,16)(17,18,23,26)(19,22,25,21)(20,27)(24,29)(28,32)(30,31);(1,3)(2,6,8,9)(4,10)(5,11)(7,13,14,15)(12,16)(17,19,22,18,23,25,21,26)(20,27,29,24,30,31,28,32);(1,4,5,12)(2,7,8,14)(3,10,11,16)(6,13,9,15)(17,20,23,30)(18,24,26,32)(19,27,25,31)(21,28,22,29)

Polycyclic presentation:

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

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 14049);