#181289

order	512
nilpotency class	3
isoclinism	isoclinic class of #7943
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄² · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	C₂×#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 ◁ C₂×C₄ ◁ C₂³×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

1460 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)(3,10)(4,8)(5,14)(6,9)(7,11)(12,15)(13,16)(17,18,20,24,22,25,21,30)(19,26,28,23,29,31,27,32);(1,3)(2,7,9,16)(4,13)(5,11)(6,12)(8,10,14,15)(17,19,21,28,22,29,20,27)(18,23,24,26,25,32,30,31);(1,4,6,5)(2,8,9,14)(3,11,12,13)(7,15,16,10);(17,20,22,21)(18,24,25,30)(19,27,29,28)(23,26,32,31)

Polycyclic presentation:

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

Conjugacy classes & character table

104 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72239624, #72239630, #72239633, …
maximal normal subgroup of	#72239624, #72239630, #72239633, …
maximal quotient of	#72239640

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 88485);