#139885

order	512
nilpotency class	4
isoclinism	isoclinic class of #139702
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂³

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#1889
commutator, prime radical	C₂²×D₄	C₂²×C₄
Frattini, cosocle	C₂×C₄×D₄	C₂³
LCS	G ▷ C₂²×D₄ ▷ C₂³ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂×C₄×D₄ ◁ G
LECS	G ▷ C₂×C₄×D₄ ▷ C₂³ ▷ C₂ ▷ C₁

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

normal subgroups only

This group has 463 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) = 128

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

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

Polycyclic presentation:

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

Conjugacy classes & character table

41 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 47081);