#139765

order	512
nilpotency class	4
isoclinism	isoclinic class of #139700
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

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

normal subgroups only

This group has 549 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) = 128

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

Polycyclic presentation:

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

Conjugacy classes & character table

53 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 46961);