#112451

order	512
nilpotency class	3
isoclinism	isoclinic class of #112446
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#1438
commutator	C₂×C₄²	C₂²×C₄
Frattini, cosocle	C₂²×C₄²	C₂³
prime radical	C₂×#532	C₂²
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂²×C₄² ▷ C₂² ▷ C₁

820 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)(3,9,11,6)(4,8)(5,7)(10,13,16,12)(14,15)(17,18,21,25,22,24,20,26)(19,27,29,23,30,31,28,32);(1,3)(2,6)(4,13)(5,11)(7,9)(8,10)(12,15)(14,16)(17,19,22,30)(18,23,24,32)(20,28,21,29)(25,31,26,27);(1,2)(3,10)(4,14)(5,7)(6,13)(8,15)(9,12)(11,16)(18,24)(19,28)(20,21)(23,31)(27,32)(29,30)

Polycyclic presentation:

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

Conjugacy classes & character table

50 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 19647);