#99594

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

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	C₂⁴	C₄×D₄
LCS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂²×C₄ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂²×C₄² ▷ C₂³ ▷ C₁

482 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 3 generators on 32 points:

(1,2,5,7,4,3,6,8)(9,10,12,15,14,11,13,16)(17,18)(19,22)(20,27,28,23)(21,24)(25,29,31,32)(26,30);(1,3,5,8,4,2,6,7)(9,11)(10,14)(12,15)(13,16)(17,19,21,26)(18,22,24,30)(20,25,28,31)(23,29,27,32);(2,3)(7,8)(9,11,14,10)(12,16,13,15)(17,20)(18,23)(19,25)(21,28)(22,29)(24,27)(26,31)(30,32)

Polycyclic presentation:

2^9|a^2=f,b^2=dfgh,c^2=g,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, 6790);