#141490

order	512
nilpotency class	4
isoclinism	isoclinic class of #141481
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂²×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#1964
commutator	C₂×C₄²	C₂²×C₄
Frattini, cosocle	C₂×C₄×Q₈	C₂³
prime radical	C₂³	#390
LCS	G ▷ C₂×C₄² ▷ C₂³ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂×C₈×Q₈ ◁ G
LECS	G ▷ C₂×C₄×Q₈ ▷ C₂³ ▷ C₂ ▷ C₁

348 conjugacy classes of subgroups; only the 78 normal subgroups are shown.

normal subgroups only

This group has 348 conjugacy classes of subgroups — too many to draw. Only its 78 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:

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

Polycyclic presentation:

2^9|a^2=f,b^2=dfg,c^2=e,d^2=ghi,e^2=gh,f^2=i,b^a=bd,c^a=ce,d^a=dg,e^a=egh,g^a=gi,h^a=hi,c^b=cg,d^b=dh,f^b=fhi,g^b=gi,h^b=hi,d^c=di,f^d=fi

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, 48686);