#140672

order	512
nilpotency class	4
isoclinism	isoclinic class of #140671
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂³

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#1964
commutator	C₂³×C₄	C₂²×C₄
Frattini, cosocle	C₂×#117	C₂³
prime radical	C₂⁴	C₄×D₄
LCS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂×#405 ◁ G
LECS	G ▷ C₂×#117 ▷ C₂³ ▷ C₂ ▷ C₁

398 conjugacy classes of subgroups; only the 84 normal subgroups are shown.

normal subgroups only

This group has 398 conjugacy classes of subgroups — too many to draw. Only its 84 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) = 128

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

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

Polycyclic presentation:

2^9|a^2=f,b^2=df,c^2=ei,e^2=i,f^2=h,b^a=bd,c^a=ce,d^a=dg,e^a=ei,g^a=gi,c^b=ch,d^b=dg,f^b=fg,g^b=gi,f^d=fi

Conjugacy classes & character table

56 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 47868);