#117356

order	512
nilpotency class	3
isoclinism	isoclinic class of #117351
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#1439
commutator	C₂×C₄²	C₂²×C₄
Frattini, cosocle	C₂²×C₄²	C₂³
prime radical	C₂⁴	#123
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂²×C₄² ▷ C₂² ▷ C₁

560 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=d,c^2=i,d^2=h,e^2=h,f^2=hi,g^2=i,b^a=bd,c^a=ce,d^a=dh,e^a=ei,f^a=fi,c^b=cf,e^b=ehi,f^b=fi,d^c=dh,e^c=eh,f^c=fhi,g^c=ghi

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 24552);