#181122

order	512
nilpotency class	3
isoclinism	isoclinic class of #7943
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄² · Schur multiplier (H₂) = C₂⁴×C₄²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	C₂×#95
commutator	C₂×C₄	C₂²×C₄²
Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂²	C₂²×#95
prime radical	C₂⁵	C₂×D₄
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂³×C₄² ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁

1636 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)(7,13)(11,16)(12,15)(17,18)(19,27,29,23)(20,24)(21,25)(22,26)(28,32,30,31);(1,3)(2,7,9,15)(4,12)(5,13)(6,11)(8,10,14,16)(17,19)(18,23)(20,30)(21,28)(22,29)(24,32)(25,31)(26,27);(3,11)(7,15)(10,16)(12,13)(17,20,22,21)(18,24,26,25)(19,28,29,30)(23,31,27,32);(1,4,6,5)(2,8,9,14)(3,12,11,13)(7,10,15,16)(17,21,22,20)(18,25,26,24)(19,28,29,30)(23,31,27,32)

Polycyclic presentation:

2^9|a^2=f,b^2=g,c^2=i,d^2=h,e^2=hi,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, #72224696, #72224697, …
maximal normal subgroup of	#72224692, #72224696, #72224697, …
maximal quotient of	#72239977, #72240803, #72257751, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 88318);