#139463

order	512
nilpotency class	4
isoclinism	isoclinic class of #139432
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	C₂×#392	C₄
LCS	G ▷ C₂³×C₄ ▷ C₂³ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂⁴ ◁ C₂²×#126 ◁ G
LECS	G ▷ C₂²×C₄⋊C₄ ▷ C₂³ ▷ C₂ ▷ C₁

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

normal subgroups only

This group has 702 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:

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

Polycyclic presentation:

2^9|a^2=f,b^2=dfgi,c^2=i,d^2=ghi,f^2=i,b^a=bd,c^a=ce,d^a=dg,e^a=egh,g^a=gi,h^a=hi,d^b=dh,e^b=egh,f^b=fhi,g^b=gi,h^b=hi,d^c=dgh,f^c=fgh,f^d=fi

Conjugacy classes & character table

47 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72235261
maximal normal subgroup of	#72235261

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 46659);