#320123

order	512
nilpotency class	3
isoclinism	isoclinic class of #299067
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂²×C₈

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂×#127
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	#2856	C₂²
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

800 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

A minimal faithful permutation representation — 4 generators on 32 points:

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

Polycyclic presentation:

2^9|a^2=fhi,b^2=eghi,c^2=hi,d^2=i,f^2=h,g^2=i,b^a=be,c^a=cf,d^a=dei,c^b=cg,d^b=dh,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, 227319);