#203109

order	512
nilpotency class	3
isoclinism	isoclinic class of #19420
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

690 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=egh,c^2=e,d^2=h,e^2=h,f^2=i,g^2=i,b^a=be,c^a=cf,d^a=di,e^a=eh,f^a=fi,c^b=ch,d^b=dhi,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, 110305);