#140133

order	512
nilpotency class	3
isoclinism	isoclinic class of #117904
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂²

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	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

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

normal subgroups only

This group has 266 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)(3,5)(4,7)(6,8)(9,10,13,18,12,17,20,24,14,11,15,19,21,16,22,23);(9,11,13,18,12,16,20,24,14,10,15,19,21,17,22,23);(2,5)(3,8)(6,7)(10,16,11,17)(13,15)(18,24,19,23)(20,22)

Polycyclic presentation:

2^9|a^2=f,b^2=df,c^2=d,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,f^b=fi,e^c=eg,f^c=fi

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 47329);