#118541

order	512
nilpotency class	3
isoclinism	isoclinic class of #118537
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#1439
commutator	C₂×C₄²	C₂²×C₄
Frattini, cosocle	C₂²×C₄²	C₂³
prime radical	C₂⁴	#121
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂²×C₄² ▷ C₂² ▷ C₁

568 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 25737);