#94863

order	512
nilpotency class	2
isoclinism	isoclinic class of H(3,ℤ/4)
automorphism	?
homology	Abelianization (H₁) = C₂×C₈² · Schur multiplier (H₂) = C₂×C₄³

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄²	C₄²
commutator	C₄	C₂×C₈²
Frattini, cosocle	C₄³	C₂³
socle	C₂³	C₂×#95
prime radical	C₂⁴	#95
LCS	G ▷ C₄ ▷ C₁
UCS	C₁ ◁ C₂×C₄² ◁ G
LECS	G ▷ C₄³ ▷ C₂³ ▷ C₁

746 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:

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

Polycyclic presentation:

2^9|a^2=e,b^2=f,c^2=h,d^2=g,e^2=h,f^2=i,b^a=bd,f^a=fg,e^b=eg

Conjugacy classes & character table

176 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 2059);