#140163

order	512
nilpotency class	3
isoclinism	isoclinic class of #117993
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	#388
commutator	C₂×C₄²	C₂²×C₄
Frattini, cosocle	C₂×C₄×C₈	C₂³
socle	C₂²	#1924
prime radical	C₂³	#433
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄×C₈ ◁ G
LECS	G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

192 conjugacy classes of subgroups; only the 68 normal subgroups are shown.

normal subgroups only

This group has 192 conjugacy classes of subgroups — too many to draw. Only its 68 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 47359);