#94564

order	512
nilpotency class	6
isoclinism	isoclinic class of #94563
automorphism	?
homology	Abelianization (H₁) = C₂×C₈ · Schur multiplier (H₂) = C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂	#7353
commutator	C₄×C₈	C₂×C₈
Frattini, cosocle	#1837	C₂²
prime radical	C₂³	#355
LCS	G ▷ C₄×C₈ ▷ C₄² ▷ C₂×C₄ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂×C₄ ◁ C₂²×C₄ ◁ C₂×C₄² ◁ #1837 ◁ G
LECS	G ▷ #1837 ▷ C₂×C₄² ▷ C₂×C₄ ▷ C₂² ▷ C₂ ▷ C₁

123 conjugacy classes of subgroups; only the 26 normal subgroups are shown.

normal subgroups only

This group has 123 conjugacy classes of subgroups — too many to draw. Only its 26 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Full subgroup data →

Presentation

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

A minimal faithful permutation representation — 2 generators on 32 points:

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

Polycyclic presentation:

2^9|a^2=d,b^2=h,c^2=g,d^2=f,e^2=h,f^2=i,g^2=i,b^a=bc,c^a=ce,e^a=eg,g^a=gh,h^a=hi,c^b=cg,d^b=deghi,e^b=eh,f^b=fi,g^b=gi,d^c=dghi,e^d=ehi,g^d=gi

Conjugacy classes & character table

32 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 1760);