#139941

order	512
nilpotency class	4
isoclinism	isoclinic class of #139712
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂²×C₄

Aliases

No aliases for this group.

Subgroups

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

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

normal subgroups only

This group has 589 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) = 64

A minimal faithful permutation representation — 3 generators on 32 points:

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

Polycyclic presentation:

2^9|a^2=f,b^2=dfgi,c^2=i,f^2=h,b^a=bd,c^a=ce,d^a=dg,e^a=eh,g^a=gi,d^b=dgi,e^b=eh,f^b=fg,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, 47137);