#141663

order	512
nilpotency class	4
isoclinism	isoclinic class of #141653
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#1970
commutator	C₄×C₈	C₂²×C₄
Frattini, cosocle	C₂×C₄×C₈	C₂³
prime radical	C₂³	#495
LCS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ #1450 ◁ G
LECS	G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

304 conjugacy classes of subgroups; only the 122 normal subgroups are shown.

normal subgroups only

This group has 304 conjugacy classes of subgroups — too many to draw. Only its 122 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) = 256

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

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

Polycyclic presentation:

2^9|a^2=f,b^2=df,c^2=eh,d^2=gi,e^2=hi,f^2=i,h^2=i,b^a=bd,c^a=ce,d^a=dg,e^a=eh,h^a=hi,d^b=di,f^b=fi

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 48859);