#139935

order	512
nilpotency class	4
isoclinism	isoclinic class of #139718
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂×C₄²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#1898
commutator	C₂³×C₄	C₂²×C₄
Frattini, cosocle	C₂²×C₄⋊C₄	C₂³
prime radical	C₂⁵	C₂²⋊C₄
LCS	G ▷ C₂³×C₄ ▷ C₂³ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂⁴ ◁ C₂²×#126 ◁ G
LECS	G ▷ C₂²×C₄⋊C₄ ▷ C₂³ ▷ C₂ ▷ C₁

558 conjugacy classes of subgroups; only the 70 normal subgroups are shown.

normal subgroups only

This group has 558 conjugacy classes of subgroups — too many to draw. Only its 70 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 — 3 generators on 32 points:

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

Polycyclic presentation:

2^9|a^2=f,b^2=dfg,c^2=i,d^2=i,f^2=hi,b^a=bd,c^a=ce,d^a=dg,e^a=eh,g^a=gi,d^b=dg,e^b=eh,f^b=fgi,g^b=gi,d^c=dh,f^c=fh,f^d=fi

Conjugacy classes & character table

47 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 47131);