#94138

order	512
nilpotency class	5
isoclinism	isoclinic class of #1404
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×Q₈	C₄×C₈
Frattini, cosocle	#2931	C₂²
socle	C₂	#7203
prime radical	#532	C₂×C₄
LCS	G ▷ C₂×Q₈ ▷ 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:

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

Polycyclic presentation:

2^9|a^2=d,b^2=e,c^2=i,d^2=g,e^2=g,f^2=i,g^2=i,b^a=bc,c^a=cf,f^a=fh,h^a=hi,c^b=ci,d^b=df,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, 1334);