#205810

order	512
nilpotency class	3
isoclinism	isoclinic class of #9781
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	#390
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂×C₄²	C₂⁴
socle	C₂²	C₂×#390
prime radical	C₂×#532	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

995 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 128

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

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

Polycyclic presentation:

2^9|a^2=g,b^2=eg,c^2=fh,d^2=hi,e^2=hi,f^2=hi,g^2=h,b^a=be,c^a=cf,e^a=eh,f^a=fhi,e^b=ei,g^b=gi

Conjugacy classes & character table

104 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 113006);