#110829

order	512
nilpotency class	3
isoclinism	isoclinic class of #1685
automorphism	?
homology	Abelianization (H₁) = C₂×C₄² · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄²	#121
commutator	C₄²	C₂×C₄²
Frattini, cosocle	C₂²×C₄²	C₂³
socle	C₂²	#1437
prime radical	#3444	C₂²
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₄² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂²×C₄² ▷ C₂² ▷ C₁

684 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 3 generators on 32 points:

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

Polycyclic presentation:

2^9|a^2=f,b^2=g,c^2=ef,d^2=h,e^2=hi,f^2=h,b^a=bd,c^a=ce,d^a=dh,e^a=ehi,g^a=ghi,c^b=ch,d^b=di

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72239292
maximal normal subgroup of	#72239292
maximal quotient of	#72239294, #72239295

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 18025);