#140095

order	512
nilpotency class	3
isoclinism	isoclinic class of #120445
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	#387
commutator	C₂×C₄²	C₂²×C₄
Frattini, cosocle	C₂×C₄×C₈	C₂³
socle	C₂²	#1917
prime radical	C₂³	#436
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄×C₈ ◁ G
LECS	G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

214 conjugacy classes of subgroups; only the 86 normal subgroups are shown.

normal subgroups only

This group has 214 conjugacy classes of subgroups — too many to draw. Only its 86 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) = 24
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 3 generators on 24 points:

(1,2,4,7)(3,5,8,6)(9,10,15,11,13,18,24,21,16,19,23,22,14,17,12,20);(9,11,15,18,13,21,24,19,16,22,23,17,14,20,12,10);(2,5)(3,8)(6,7)(9,12,13,15,16,24,14,23)(10,11,17,20,19,22,18,21)

Polycyclic presentation:

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

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 47291);