#139845

order	512
nilpotency class	4
isoclinism	isoclinic class of #139844
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

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

normal subgroups only

This group has 544 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:

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

Polycyclic presentation:

2^9|a^2=f,b^2=dfh,e^2=i,f^2=i,b^a=bd,c^a=ce,d^a=dg,e^a=ehi,g^a=gi,h^a=hi,d^b=dgi,e^b=eh,f^b=fg,g^b=gi,h^b=hi,d^c=dh,e^c=ei,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, 47041);