#140823

order	512
nilpotency class	3
isoclinism	isoclinic class of #113366
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	#390
commutator	C₂×C₄²	C₂²×C₄
Frattini, cosocle	C₂×C₄×C₈	C₂³
socle	C₂²	#1955
prime radical	C₂²×D₄	M₁₆
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄×C₈ ◁ G
LECS	G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

269 conjugacy classes of subgroups; only the 76 normal subgroups are shown.

normal subgroups only

This group has 269 conjugacy classes of subgroups — too many to draw. Only its 76 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) = 128

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

Polycyclic presentation:

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

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, 48019);