#99593

order	512
nilpotency class	3
isoclinism	isoclinic class of #9914
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂³

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂²×C₄	#120
commutator	C₂³×C₄	C₂²×C₄
Frattini, cosocle	C₂²×C₄²	C₂³
socle	C₂³	#386
prime radical	C₂×#121	C₂×C₄
LCS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂²×C₄ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂²×C₄² ▷ C₂³ ▷ C₁

533 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 24
transitive degree	μ_t(G) = 256

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

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

Polycyclic presentation:

2^9|a^2=f,b^2=dfgh,d^2=g,e^2=g,f^2=i,b^a=bd,c^a=ce,e^a=egh,c^b=ci,d^b=dg,e^b=eh,f^b=fg,d^c=dh,e^c=eg,f^c=fh

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, 6789);