#95557

order	512
nilpotency class	3
isoclinism	isoclinic class of #1284
automorphism	?
homology	Abelianization (H₁) = C₂×C₄×C₈ · Schur multiplier (H₂) = C₂×C₄²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂²×C₄	#95
commutator	C₂×C₄	C₂×C₄×C₈
Frattini, cosocle	C₄³	C₂³
socle	C₂³	C₂×#95
prime radical	C₂⁴	#95
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₄ ◁ C₂×C₄³ ◁ G
LECS	G ▷ C₄³ ▷ C₂³ ▷ C₁

608 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)(3,7)(4,6)(5,8)(9,10,12,15,14,16,13,11)(17,18)(19,25,26,22)(20,24)(21,23)(27,31,32,29)(28,30);(2,6,8,7)(9,11,13,16,14,15,12,10)(17,19,20,27,21,26,28,32)(18,22,24,29,23,25,30,31);(1,3,5,4)(2,7,8,6)(9,12,14,13)(10,15,16,11)

Polycyclic presentation:

2^9|a^2=e,b^2=f,c^2=ghi,d^2=gh,e^2=i,f^2=hi,b^a=bd,d^a=dg,f^a=fgh,e^b=eh

Conjugacy classes & character table

128 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 2753);