#94641

order	512
nilpotency class	6
isoclinism	isoclinic class of #94638
automorphism	?
homology	Abelianization (H₁) = C₂×C₄ · Schur multiplier (H₂) = C₂²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂	#7402
commutator	C₄³	C₂×C₄
Frattini, cosocle	#1962	C₂²
prime radical	#532	D₄
LCS	G ▷ C₄³ ▷ C₂×C₄² ▷ C₂×C₄ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂×C₄ ◁ C₂²×C₄ ◁ C₂×C₄² ◁ #1962 ◁ G
LECS	G ▷ #1962 ▷ C₂×C₄² ▷ C₂×C₄ ▷ C₂² ▷ C₂ ▷ C₁

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

normal subgroups only

This group has 194 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:

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

Polycyclic presentation:

2^9|a^2=d,b^2=i,c^2=f,d^2=i,e^2=h,g^2=i,b^a=bc,c^a=ce,e^a=eg,f^a=fh,g^a=gh,h^a=hi,c^b=cf,d^b=defh,e^b=eh,g^b=gi,d^c=dghi,e^d=ehi,f^d=fi,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, 1837);