#139654

order	512
nilpotency class	4
isoclinism	isoclinic class of #139598
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂²×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#1889
commutator	C₂×C₂²⋊C₄	C₂²×C₄
Frattini, cosocle	C₂²×C₂²⋊C₄	C₂³
prime radical	#3490	C₂²
LCS	G ▷ C₂×C₂²⋊C₄ ▷ C₂³ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂⁴ ◁ C₂²×C₂²⋊C₄ ◁ G
LECS	G ▷ C₂²×C₂²⋊C₄ ▷ C₂³ ▷ C₂ ▷ C₁

668 conjugacy classes of subgroups; only the 68 normal subgroups are shown.

normal subgroups only

This group has 668 conjugacy classes of subgroups — too many to draw. Only its 68 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

47 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72237944, #72238389, #72238393
maximal normal subgroup of	#72237944, #72238389, #72238393

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 46850);