#106727

order	512
nilpotency class	3
isoclinism	isoclinic class of #8359
automorphism	?
homology	Abelianization (H₁) = C₂×C₄² · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	#379
commutator	C₂²×C₄	C₂×C₄²
Frattini, cosocle	C₂²×C₄²	C₂³
socle	C₂²	C₂²⋊#95
prime radical	C₂²×#41	Q₈
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂²×C₄² ▷ C₂² ▷ C₁

819 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 13923);