#139704

order	512
nilpotency class	4
isoclinism	isoclinic class of #139700
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂⁵

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	#23902	C₂
LCS	G ▷ C₂²×D₄ ▷ C₂³ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂²×#41 ◁ G
LECS	G ▷ C₂²×#41 ▷ C₂³ ▷ C₂ ▷ C₁

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

normal subgroups only

This group has 753 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) = 40
transitive degree	μ_t(G) = 64

A minimal faithful permutation representation — 3 generators on 40 points:

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

Polycyclic presentation:

2^9|a^2=f,b^2=df,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, 46900);