#121561

order	512
nilpotency class	3
isoclinism	isoclinic class of #10176
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂³

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	#392
commutator	C₂×C₄²	C₂²×C₄
Frattini, cosocle	C₂²×C₄²	C₂³
socle	C₂²	#1444
prime radical	C₂⁴	#125
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂²×C₄² ▷ C₂² ▷ C₁

464 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 128

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

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

Polycyclic presentation:

2^9|a^2=g,b^2=d,c^2=e,d^2=h,e^2=hi,f^2=h,g^2=i,b^a=bd,c^a=ce,d^a=dh,e^a=ehi,f^a=fi,c^b=cf,f^b=fhi,d^c=di,f^c=fh

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 28757);