#197303

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	#60916	C₂
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

1229 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=eg,e^2=hi,f^2=h,g^2=hi,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, 104499);