#141555

order	512
nilpotency class	4
isoclinism	isoclinic class of #141503
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#1972
commutator	C₂²×C₈	C₂²×C₄
Frattini, cosocle	C₂×C₄×C₈	C₂³
prime radical	C₂×#2250	C₂
LCS	G ▷ C₂²×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ #1451 ◁ G
LECS	G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

366 conjugacy classes of subgroups; only the 80 normal subgroups are shown.

normal subgroups only

This group has 366 conjugacy classes of subgroups — too many to draw. Only its 80 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:

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

Polycyclic presentation:

2^9|a^2=f,b^2=dfgh,c^2=eh,d^2=ghi,e^2=gh,f^2=h,g^2=i,b^a=bd,c^a=ce,d^a=dg,e^a=egh,g^a=gi,c^b=cg,d^b=dh,e^b=ei,f^b=fh,f^c=fi

Conjugacy classes & character table

44 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 48751);