#181191

order	512
nilpotency class	3
isoclinism	isoclinic class of #1284
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄² · Schur multiplier (H₂) = C₂⁵×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄²	#95
commutator	C₂×C₄	C₂²×C₄²
Frattini, cosocle	C₂×C₄²	C₂⁴
socle	C₂²	C₂²×#95
prime radical	C₂⁵	C₄⋊C₄
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄² ◁ C₂³×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

1530 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 64

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

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

Polycyclic presentation:

2^9|a^2=f,b^2=g,c^2=i,d^2=h,e^2=hi,f^2=h,g^2=h,b^a=be,e^a=eh,g^a=ghi,f^b=fi

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, 88387);