#110682

order	512
nilpotency class	3
isoclinism	isoclinic class of #9369
automorphism	?
homology	Abelianization (H₁) = C₂×C₄² · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	#386
commutator	C₄²	C₂×C₄²
Frattini, cosocle	C₂²×C₄²	C₂³
socle	C₂²	#1437
prime radical	C₂×#532	C₂²
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂²×C₄² ▷ C₂² ▷ C₁

673 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 64

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

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

Polycyclic presentation:

2^9|a^2=f,b^2=g,c^2=efh,d^2=i,e^2=h,g^2=hi,b^a=bd,c^a=ce,d^a=dh,e^a=eh,d^b=di,e^b=ehi,f^b=fhi,d^c=dhi

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72239081
maximal normal subgroup of	#72239081
maximal quotient of	#72239116

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 17878);