#139769

order	512
nilpotency class	4
isoclinism	isoclinic class of #139706
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	C₂²×D₄	C₂²×C₄
Frattini, cosocle	C₂²×#41	C₂³
prime radical	#3485	C₂²
LCS	G ▷ C₂²×D₄ ▷ C₂³ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂²×#41 ◁ G
LECS	G ▷ C₂²×#41 ▷ C₂³ ▷ C₂ ▷ C₁

523 conjugacy classes of subgroups; only the 64 normal subgroups are shown.

normal subgroups only

This group has 523 conjugacy classes of subgroups — too many to draw. Only its 64 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) = 64

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

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

Polycyclic presentation:

2^9|a^2=f,b^2=df,c^2=g,e^2=i,f^2=i,b^a=bd,c^a=ce,d^a=dg,e^a=ehi,h^a=hi,d^b=dg,e^b=ehi,f^b=fg,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, 46965);