#141159

order	512
nilpotency class	4
isoclinism	isoclinic class of #140953
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂×C₄

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

Polycyclic presentation:

2^9|a^2=f,b^2=df,c^2=e,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, 48355);