#203066

order	512
nilpotency class	3
isoclinism	isoclinic class of #19186
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	C₂×#121
commutator	C₄²	C₂³×C₄
Frattini, cosocle	C₂×C₄²	C₂⁴
socle	C₂²	C₂×#388
prime radical	C₂×#532	C₂²
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

733 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 4 generators on 32 points:

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

Polycyclic presentation:

2^9|a^2=g,b^2=egh,c^2=e,e^2=h,f^2=i,g^2=i,b^a=be,c^a=cf,e^a=eh,f^a=fi,d^c=dh,f^c=fhi

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 110262);