#141461

order	512
nilpotency class	4
isoclinism	isoclinic class of #141454
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂²×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#1970
commutator	C₂²×C₈	C₂²×C₄
Frattini, cosocle	C₂³×C₈	C₂³
prime radical	C₂×#2215	C₂
LCS	G ▷ C₂²×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ C₈×C₂²⋊C₄ ◁ G
LECS	G ▷ C₂³×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

367 conjugacy classes of subgroups; only the 82 normal subgroups are shown.

normal subgroups only

This group has 367 conjugacy classes of subgroups — too many to draw. Only its 82 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)(3,5,7,8)(4,6)(9,10,15,21,16,23,32,22)(11,18,26,38,27,35,40,39)(12,28,24,17,30,36,25,37)(13,31,33,19,14,34,29,20);(1,3,4,7)(2,5)(6,8)(9,11,13,24,32,40,29,12,16,27,14,25,15,26,33,30)(10,17,20,35,21,36,31,39,23,37,19,18,22,28,34,38);(2,6)(5,8)(9,12,29,26,15,24,13,27,16,30,33,40,32,25,14,11)(10,18,34,37,22,39,19,36,23,35,31,17,21,38,20,28)

Polycyclic presentation:

2^9|a^2=f,b^2=dfgi,c^2=eg,d^2=ghi,e^2=ghi,f^2=i,g^2=i,b^a=bd,c^a=ce,d^a=dg,e^a=egh,g^a=gi,c^b=ci,d^b=dh,f^b=fh

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 48657);