#197628

order	512
nilpotency class	3
isoclinism	isoclinic class of #19552
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁶

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×#386
prime radical	#60900	C₂
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

1088 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)(3,10)(4,8)(5,14)(6,9)(7,12)(11,15)(13,16)(17,18)(19,23,28,32)(20,30)(21,25)(22,26)(24,29,31,27);(1,3,5,12,6,13,4,11)(2,7,8,10,9,15,14,16)(17,19)(18,23,26,32)(20,29)(21,27)(22,28)(24,30,31,25);(1,2)(3,7)(4,14)(5,8)(6,9)(10,12)(11,16)(13,15)(17,19)(18,24)(20,29)(21,27)(22,28)(23,30)(25,32)(26,31);(1,4,6,5)(2,8,9,14)(3,11,13,12)(7,10,15,16)(17,20,22,21)(18,25,26,30)(19,27,28,29)(23,31,32,24)

Polycyclic presentation:

2^9|a^2=g,b^2=eg,d^2=h,e^2=hi,f^2=i,b^a=be,c^a=cf,d^a=di,e^a=eh,d^b=di,e^b=ei,f^b=fi,g^b=gi,d^c=dh,e^c=ei,f^c=fi,g^c=gi

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 104824);