#197690

order	512
nilpotency class	3
isoclinism	isoclinic class of #19504
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	#60946	C₂
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

1085 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:

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

Polycyclic presentation:

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

74 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 104886);