#139922

order	512
nilpotency class	4
isoclinism	isoclinic class of #139724
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂³

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#1898
commutator	C₂³×C₄	C₂²×C₄
Frattini, cosocle	C₂×#117	C₂³
prime radical	C₂×#540	C₂²
LCS	G ▷ C₂³×C₄ ▷ C₂³ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂×#540 ◁ G
LECS	G ▷ C₂×#117 ▷ C₂³ ▷ C₂ ▷ C₁

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

normal subgroups only

This group has 483 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) = 48
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 3 generators on 48 points:

(1,2,5,9,6,8,15,3)(4,11,13,16,12,7,14,10)(17,18)(19,28,32,44)(20,33,29,25)(21,24)(22,27)(23,30)(26,36)(31,42,47,43)(34,39)(35,41,45,40)(37,46)(38,48);(1,3,5,8,6,9,15,2)(4,10,14,11,12,16,13,7)(17,19,29,30,22,32,20,34)(18,23,33,44,27,39,25,28)(21,31,45,46,36,47,35,48)(24,37,41,43,26,38,40,42);(1,4)(2,7)(3,10)(5,14)(6,12)(8,11)(9,16)(13,15)(18,24)(21,36)(23,37)(25,40)(26,27)(28,42)(31,47)(33,41)(35,45)(38,39)(43,44)(46,48)

Polycyclic presentation:

2^9|a^2=f,b^2=df,e^2=i,f^2=h,b^a=bd,c^a=ce,d^a=dg,e^a=ehi,g^a=gi,d^b=dg,e^b=eh,f^b=fg,g^b=gi,d^c=dh,e^c=ei,f^c=fh,f^d=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, 47118);