#131679

order	512
nilpotency class	4
isoclinism	isoclinic class of #131677
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#1786
commutator	C₄×C₈	C₂²×C₄
Frattini, cosocle	C₂×C₄×C₈	C₂³
prime radical	#62425	C₂
LCS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ #2965 ◁ G
LECS	G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

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

normal subgroups only

This group has 612 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) = 24
transitive degree	μ_t(G) = 128

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

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

Polycyclic presentation:

2^9|a^2=f,b^2=i,d^2=hi,e^2=gi,f^2=hi,h^2=i,b^a=bd,c^a=ce,d^a=dg,e^a=ei,d^b=dh,e^b=ei,f^b=fgh,h^b=hi,d^c=di,e^c=egi,f^c=fg

Conjugacy classes & character table

47 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 38875);