#181222

order	512
nilpotency class	3
isoclinism	isoclinic class of #7943
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄² · Schur multiplier (H₂) = C₂⁵×C₄

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	C₂²⋊C₄
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂³×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

1548 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)(12,16)(13,15)(17,18)(19,26,28,32)(20,24)(21,30)(22,25)(23,27,31,29);(1,3)(2,7,9,16)(4,12)(5,11)(6,13)(8,10,14,15)(17,19,21,29,22,28,20,27)(18,23,24,26,25,31,30,32);(1,4,6,5)(2,8,9,14)(3,11,13,12)(7,15,16,10);(1,4,6,5)(2,8,9,14)(3,12,13,11)(7,10,16,15)(17,20,22,21)(18,24,25,30)(19,27,28,29)(23,26,31,32)

Polycyclic presentation:

2^9|a^2=f,b^2=g,c^2=h,d^2=hi,e^2=hi,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	#72224692, #72224699, #72224700, …
maximal normal subgroup of	#72224692, #72224699, #72224700, …
maximal quotient of	#72257629

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 88418);