#139840

order	512
nilpotency class	4
isoclinism	isoclinic class of #139840
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂	#9499
commutator	C₂³×C₄	C₂²×C₄
Frattini, cosocle	C₂³×D₄	C₂³
prime radical	#3484	C₂²
LCS	G ▷ C₂³×C₄ ▷ C₂³ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂³ ◁ C₂³×D₄ ◁ G
LECS	G ▷ C₂³×D₄ ▷ C₂³ ▷ C₂ ▷ C₁

601 conjugacy classes of subgroups; only the 66 normal subgroups are shown.

normal subgroups only

This group has 601 conjugacy classes of subgroups — too many to draw. Only its 66 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)(3,12)(4,17)(5,20,6,10)(7,11)(8,14,23,15)(9,18,24,21)(13,22,30,32)(16,27)(19,25)(26,29)(28,31);(1,3,6,14)(2,8,10,12)(4,13,21,29)(5,15,7,16)(9,22,25,26)(11,23,20,27)(17,31,24,32)(18,28,19,30);(1,4)(2,9)(3,13)(5,18)(6,21)(7,19)(8,22)(10,25)(11,24)(12,26)(14,29)(15,28)(16,30)(17,20)(23,32)(27,31)

Polycyclic presentation:

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

Conjugacy classes & character table

38 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72338684, #72338776, #72338783
maximal normal subgroup of	#72338684, #72338776, #72338783

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 47036);