#99533

order	512
nilpotency class	3
isoclinism	isoclinic class of #10046
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	C₂×C₄²	C₂²×C₄
Frattini, cosocle	C₄³	C₂³
socle, prime radical	C₂³	#386
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂²×C₄ ◁ C₄³ ◁ G
LECS	G ▷ C₄³ ▷ C₂³ ▷ C₁

462 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 512

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

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

Polycyclic presentation:

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

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6729);