#107338

order	512
nilpotency class	3
isoclinism	isoclinic class of #107334
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	C₄²	C₂×C₄²
Frattini, cosocle	C₂²×C₄²	C₂³
prime radical	#3444	C₂²
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂²×C₄² ▷ C₂² ▷ C₁

759 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:

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

Polycyclic presentation:

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

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, 14534);