#94673

order	512
nilpotency class	6
isoclinism	isoclinic class of #94670
automorphism	?
homology	Abelianization (H₁) = C₂×C₄ · Schur multiplier (H₂) = C₂²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#1402
commutator	#384	C₂×C₄
Frattini, cosocle	#1956	C₂²
prime radical	C₂³	AΣL(1,𝔽₁₆)
LCS	G ▷ #384 ▷ C₂×C₄² ▷ C₂×C₄ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³ ◁ C₂²×C₄ ◁ C₂×C₄² ◁ #1956 ◁ G
LECS	G ▷ #1956 ▷ C₂×C₄² ▷ C₂×C₄ ▷ C₂² ▷ C₂ ▷ C₁

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

normal subgroups only

This group has 189 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) = 40
transitive degree	μ_t(G) = 64

A minimal faithful permutation representation — 2 generators on 40 points:

(1,2)(3,5)(4,7)(6,8)(9,10,12,17,14,19,26,35)(11,20,21,33,23,30,37,40)(13,16,24,32,27,18,25,34)(15,28,29,38,31,36,39,22);(2,5)(3,8)(6,7)(9,11,14,23)(10,15,19,31)(12,21,26,37)(13,22,27,38)(16,20,18,30)(17,33,35,40)(24,28,25,36)(29,32,39,34)

Polycyclic presentation:

2^9|a^2=d,b^2=i,c^2=f,d^2=i,e^2=hi,g^2=i,b^a=bc,c^a=ce,e^a=eg,g^a=gh,h^a=hi,c^b=cf,d^b=defhi,e^b=ehi,g^b=ghi,d^c=dgh,e^c=ehi,g^c=gi,e^d=ehi,g^d=gi

Conjugacy classes & character table

41 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 1869);