#107730

order	512
nilpotency class	3
isoclinism	isoclinic class of #107718
automorphism	?
homology	Abelianization (H₁) = C₂×C₄² · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂²⋊#95
commutator	C₂²×C₄	C₂×C₄²
Frattini, cosocle	C₂²×C₄²	C₂³
prime radical	#3444	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂²×C₄² ▷ C₂² ▷ C₁

748 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 14926);