#139839

order	512
nilpotency class	4
isoclinism	isoclinic class of #139838
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	C₂²×#41	C₂³
prime radical	#3485	C₂²
LCS	G ▷ C₂³×C₄ ▷ C₂³ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂³ ◁ C₂²×#41 ◁ G
LECS	G ▷ C₂²×#41 ▷ C₂³ ▷ C₂ ▷ C₁

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

normal subgroups only

This group has 555 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:

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

Polycyclic presentation:

2^9|a^2=f,b^2=dfh,d^2=i,f^2=i,b^a=bd,c^a=ce,d^a=dg,e^a=eh,g^a=gi,h^a=hi,d^b=dg,e^b=eh,f^b=fgi,g^b=gi,h^b=hi,d^c=dh,f^c=fh,g^c=gi,f^d=fi,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, 47035);