#181126

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

1658 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)(19,25,27,22)(20,23)(21,24)(26,31,28,30)(29,32);(1,3)(2,7,9,16)(4,12)(5,11)(6,13)(8,10,14,15)(17,19)(18,22)(20,28)(21,27)(23,31)(24,25)(26,29)(30,32);(1,4,6,5)(2,8,9,14)(3,11,13,12)(7,15,16,10)(17,20,21,29)(18,23,24,32)(19,26,27,28)(22,30,25,31);(1,4,6,5)(2,8,9,14)(3,12,13,11)(7,10,16,15)

Polycyclic presentation:

2^9|a^2=f,b^2=g,c^2=h,d^2=hi,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	#72239002, #72239006, #72239010, …
maximal normal subgroup of	#72239002, #72239006, #72239010, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 88322);