#94137

order	512
nilpotency class	5
isoclinism	isoclinic class of #1402
automorphism	?
homology	Abelianization (H₁) = C₄×C₈ · Schur multiplier (H₂) = C₂²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₈	AΣL(1,𝔽₁₆)
commutator	C₂×D₄	C₄×C₈
Frattini, cosocle	#2931	C₂²
socle	C₂	#7203
prime radical	#532	C₂×C₄
LCS	G ▷ C₂×D₄ ▷ C₂×C₄ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₈ ◁ C₂×C₈ ◁ C₂²×C₈ ◁ #2931 ◁ G
LECS	G ▷ #2931 ▷ C₂²×C₄ ▷ C₂² ▷ C₂ ▷ C₁

207 conjugacy classes of subgroups; only the 52 normal subgroups are shown.

normal subgroups only

This group has 207 conjugacy classes of subgroups — too many to draw. Only its 52 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:

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

Polycyclic presentation:

2^9|a^2=d,b^2=e,d^2=g,e^2=g,f^2=i,g^2=i,b^a=bc,c^a=cf,f^a=fh,h^a=hi,d^b=dfi,f^b=fi,h^b=hi,d^c=dhi,f^c=fi,f^d=fi

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 1333);