#205809

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	C₂⁴	C₄×D₄
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

925 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=egh,c^2=f,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, 113005);