#106904

order	512
nilpotency class	3
isoclinism	isoclinic class of #106878
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	C₂⁴	C₄×D₄
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂²×C₄² ▷ C₂² ▷ C₁

772 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:

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

Polycyclic presentation:

2^9|a^2=f,b^2=g,c^2=h,e^2=h,f^2=hi,g^2=i,b^a=bd,c^a=ce,d^a=dh,g^a=gi,d^b=di,f^b=fh,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, 14100);