#141161

order	512
nilpotency class	4
isoclinism	isoclinic class of #140953
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₄²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#1964
commutator	C₂×C₄²	C₂²×C₄
Frattini, cosocle	C₂×#117	C₂³
prime radical	C₂⁴	#121
LCS	G ▷ C₂×C₄² ▷ C₂³ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂×#405 ◁ G
LECS	G ▷ C₂×#117 ▷ C₂³ ▷ C₂ ▷ C₁

366 conjugacy classes of subgroups; only the 88 normal subgroups are shown.

normal subgroups only

This group has 366 conjugacy classes of subgroups — too many to draw. Only its 88 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:

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

Polycyclic presentation:

2^9|a^2=f,b^2=dfi,c^2=ei,d^2=gh,e^2=i,f^2=gh,b^a=bd,c^a=ce,d^a=dg,e^a=ei,g^a=gi,h^a=hi,c^b=cg,d^b=dh,f^b=fh,g^b=gi,h^b=hi,d^c=di,f^d=fi

Conjugacy classes & character table

50 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 48357);