#140808

order	512
nilpotency class	3
isoclinism	isoclinic class of #1685
automorphism	?
homology	Abelianization (H₁) = C₂²×C₈ · Schur multiplier (H₂) = C₂×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₈	#121
commutator	C₄²	C₂²×C₈
Frattini, cosocle	C₂×C₄×C₈	C₂³
socle	C₂²	#1954
prime radical	#3169	C₄
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₈ ◁ C₂×C₄×C₈ ◁ G
LECS	G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

301 conjugacy classes of subgroups; only the 94 normal subgroups are shown.

normal subgroups only

This group has 301 conjugacy classes of subgroups — too many to draw. Only its 94 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) = 48
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 3 generators on 48 points:

(1,2,6,10,7,9,5,3)(4,12,14,16,15,11,13,8)(17,18,21,26,22,27,36,44,23,28,37,45,20,25,30,39)(19,29,33,46,32,40,47,43,31,42,35,48,34,24,38,41);(1,3,6,9,7,10,5,2)(4,11,14,16,15,12,13,8)(17,19,30,47,20,34,37,33,23,31,36,38,22,32,21,35)(18,24,39,46,25,42,45,41,28,40,44,48,27,29,26,43);(1,4,6,14,7,15,5,13)(2,8,3,11,9,16,10,12)(19,31)(24,40)(29,42)(32,34)(33,35)(38,47)(41,43)(46,48)

Polycyclic presentation:

2^9|a^2=f,b^2=df,c^2=eg,d^2=i,e^2=g,f^2=h,h^2=i,b^a=bd,c^a=ce,d^a=dg,e^a=eg,c^b=ci,d^b=dgi,f^b=fgi

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, 48004);