#112668

order	512
nilpotency class	3
isoclinism	isoclinic class of #10128
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	#392
commutator	C₂×C₄²	C₂²×C₄
Frattini, cosocle	C₂²×C₄²	C₂³
socle	C₂²	#1438
prime radical	C₂⁴	#121
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂²×C₄² ▷ C₂² ▷ C₁

806 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)(3,9,12,6)(4,13)(5,7)(8,15)(10,11,16,14)(17,18,22,28,23,27,21,26)(19,25,30,24,20,31,29,32);(1,3)(2,6)(4,14)(5,12)(7,9)(8,10)(11,15)(13,16)(17,19,23,20)(18,24,27,32)(21,30,22,29)(25,26,31,28);(1,2)(3,10)(4,13)(5,7)(6,14)(8,15)(9,11)(12,16)(17,20,23,19)(18,25,27,31)(21,29,22,30)(24,28,32,26)

Polycyclic presentation:

2^9|a^2=g,b^2=i,c^2=i,d^2=i,f^2=h,g^2=i,b^a=bd,c^a=ce,d^a=dh,e^a=ei,f^a=fhi,c^b=cf,d^b=di,f^b=fh,g^b=ghi,d^c=dhi,f^c=fh,g^c=gi

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, 19864);