#94545

order	512
nilpotency class	6
isoclinism	isoclinic class of #7523
automorphism	?
homology	Abelianization (H₁) = C₂×C₈ · Schur multiplier (H₂) = C₂²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#1404
commutator	#125	C₂×C₈
Frattini, cosocle	#1992	C₂²
prime radical	C₂³	AΣL(1,𝔽₁₆)
LCS	G ▷ #125 ▷ C₄² ▷ C₂×C₄ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³ ◁ C₂²×C₄ ◁ C₂×C₄² ◁ #1992 ◁ G
LECS	G ▷ #1992 ▷ C₂×C₄² ▷ C₂×C₄ ▷ C₂² ▷ C₂ ▷ C₁

200 conjugacy classes of subgroups; only the 28 normal subgroups are shown.

normal subgroups only

This group has 200 conjugacy classes of subgroups — too many to draw. Only its 28 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:

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

Polycyclic presentation:

2^9|a^2=d,b^2=i,c^2=h,d^2=f,e^2=hi,g^2=i,b^a=bc,c^a=ce,e^a=eg,g^a=gh,h^a=hi,c^b=ch,d^b=dei,e^b=ehi,g^b=gh,d^c=dgh,e^c=eh,g^c=gi,e^d=ehi,g^d=gi

Conjugacy classes & character table

38 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 1741);