#139733

order	512
nilpotency class	4
isoclinism	isoclinic class of #139733
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂	#9495
commutator	C₂²×D₄	C₂²×C₄
Frattini, cosocle	C₂³×D₄	C₂³
prime radical	#23912	C₂
LCS	G ▷ C₂²×D₄ ▷ C₂³ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂³ ◁ C₂³×D₄ ◁ G
LECS	G ▷ C₂³×D₄ ▷ C₂³ ▷ C₂ ▷ C₁

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

normal subgroups only

This group has 724 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)(3,13,16,25)(4,17,20,26)(5,11,6,10)(7,12)(8,15)(9,19)(14,22)(18,24)(21,23)(27,30,31,28)(29,32);(1,3,6,15,7,16,5,21)(2,8,11,25,12,23,10,13)(4,14,18,30,20,29,19,28)(9,22,26,31,24,32,17,27);(1,4)(2,9)(3,14)(5,19)(6,18)(7,20)(8,22)(10,17)(11,26)(12,24)(13,27)(15,30)(16,29)(21,28)(23,32)(25,31)

Polycyclic presentation:

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

Conjugacy classes & character table

38 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 46929);