#140057

order	512
nilpotency class	4
isoclinism	isoclinic class of #139970
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂⁴

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

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

normal subgroups only

This group has 610 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:

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

Polycyclic presentation:

2^9|a^2=f,b^2=df,c^2=gi,d^2=gi,e^2=g,f^2=gi,b^a=bd,c^a=ce,d^a=dg,e^a=egh,h^a=hi,d^b=di,e^b=ehi,f^b=fi,d^c=dh,e^c=eg,f^c=fh,e^d=ei,f^e=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, 47253);