#203053

order	512
nilpotency class	3
isoclinism	isoclinic class of #19504
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	C₂⁴	#123
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

651 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=egi,c^2=e,e^2=h,f^2=i,g^2=h,b^a=be,c^a=cf,d^a=di,e^a=eh,f^a=fi,d^b=dhi,d^c=dh,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, 110249);