#119755

order	512
nilpotency class	3
isoclinism	isoclinic class of #10500
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	#394
commutator	C₂×C₄²	C₂²×C₄
Frattini, cosocle	C₂²×C₄²	C₂³
socle	C₂²	#1440
prime radical	C₂×#532	C₂²
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂²×C₄² ▷ C₂² ▷ C₁

634 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:

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

Polycyclic presentation:

2^9|a^2=g,c^2=d,d^2=h,e^2=i,f^2=h,g^2=i,b^a=bd,c^a=ce,d^a=dh,e^a=ei,f^a=fi,c^b=cf,d^b=dh,e^b=ei,f^b=fh,e^c=ehi

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, 26951);