#106343

order	512
nilpotency class	3
isoclinism	isoclinic class of #106326
automorphism	?
homology	Abelianization (H₁) = C₂×C₄² · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂²⋊#95
commutator	C₂²×C₄	C₂×C₄²
Frattini, cosocle	C₂²×C₄²	C₂³
prime radical	C₂²×#41	Q₈
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂²×C₄² ▷ C₂² ▷ C₁

795 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:

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

Polycyclic presentation:

2^9|a^2=f,b^2=g,d^2=hi,f^2=i,g^2=hi,b^a=bd,c^a=ce,d^a=dh,g^a=ghi,e^b=eh,f^b=fi,d^c=dh

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, 13539);