#141191

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

371 conjugacy classes of subgroups; only the 80 normal subgroups are shown.

normal subgroups only

This group has 371 conjugacy classes of subgroups — too many to draw. Only its 80 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:

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

Polycyclic presentation:

2^9|a^2=f,b^2=dfi,c^2=eh,d^2=gh,e^2=i,f^2=gh,b^a=bd,c^a=ce,d^a=dg,g^a=gi,h^a=hi,c^b=cg,d^b=dh,e^b=ei,f^b=fh,g^b=gi,h^b=hi,f^c=fi,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, 48387);