#197109

order	512
nilpotency class	3
isoclinism	isoclinic class of #197038
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×#386
commutator	C₄²	C₂³×C₄
Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	C₂×#532	C₂²
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

1095 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 64

A minimal faithful permutation representation — 4 generators on 32 points:

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

Polycyclic presentation:

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

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, 104305);