#94553

order	512
nilpotency class	6
isoclinism	isoclinic class of #94550
automorphism	?
homology	Abelianization (H₁) = C₂×C₈ · Schur multiplier (H₂) = C₂

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂	#7348
commutator	C₂×C₄²	C₂×C₈
Frattini, cosocle	#1822	C₂²
prime radical	C₂⁴	#100
LCS	G ▷ C₂×C₄² ▷ C₂²×C₄ ▷ C₂³ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂² ◁ C₂²×C₄ ◁ C₂³×C₄ ◁ #1822 ◁ G
LECS	G ▷ #1822 ▷ C₂³×C₄ ▷ C₂³ ▷ C₂² ▷ C₂ ▷ C₁

130 conjugacy classes of subgroups; only the 24 normal subgroups are shown.

normal subgroups only

This group has 130 conjugacy classes of subgroups — too many to draw. Only its 24 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:

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

Polycyclic presentation:

2^9|a^2=d,b^2=gi,c^2=h,d^2=f,e^2=i,f^2=i,b^a=bc,c^a=ce,e^a=eg,g^a=gh,h^a=hi,d^b=dehi,f^b=fhi,d^c=dgi,f^c=fi,e^d=eh,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, 1749);