#106189

order	512
nilpotency class	3
isoclinism	isoclinic class of #106181
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	#117
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) = 128

A minimal faithful permutation representation — 3 generators on 32 points:

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

Polycyclic presentation:

2^9|a^2=f,b^2=g,c^2=i,e^2=h,f^2=hi,g^2=h,b^a=bd,c^a=ce,d^a=dh,e^a=ei,g^a=gh,d^b=dh,e^b=eh,f^b=fh,d^c=dh,e^c=eh,f^c=fhi

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 13385);