#139650

order	512
nilpotency class	4
isoclinism	isoclinic class of #139604
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂²×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#1889
commutator	C₂×C₄⋊C₄	C₂²×C₄
Frattini, cosocle	C₂²×C₄⋊C₄	C₂³
prime radical	#3492	C₂²
LCS	G ▷ C₂×C₄⋊C₄ ▷ C₂³ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂⁴ ◁ C₂²×C₄⋊C₄ ◁ G
LECS	G ▷ C₂²×C₄⋊C₄ ▷ C₂³ ▷ C₂ ▷ C₁

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

normal subgroups only

This group has 603 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) = 32

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

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

Polycyclic presentation:

2^9|a^2=f,b^2=df,c^2=i,d^2=g,e^2=i,f^2=g,b^a=bd,c^a=ce,d^a=dg,e^a=ehi,h^a=hi,c^b=cgi,e^b=ehi,d^c=dh,e^c=ei,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, 46846);