#199162

order	512
nilpotency class	3
isoclinism	isoclinic class of #198843
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	#60946	C₂
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

1180 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)(3,6,10,14)(4,7)(5,8)(9,12)(11,13,16,15)(17,18,21,23,22,26,20,25)(19,27,28,24,30,31,29,32);(1,3)(2,6,8,14)(4,11)(5,10)(7,13,9,15)(12,16)(18,23)(20,21)(24,27)(25,26)(28,29)(31,32);(2,7)(4,12)(6,13)(8,9)(11,16)(14,15)(17,19)(18,24)(20,28)(21,29)(22,30)(23,27)(25,31)(26,32);(2,8)(6,14)(7,9)(13,15)(17,20,22,21)(18,23,26,25)(19,28,30,29)(24,27,32,31)

Polycyclic presentation:

2^9|a^2=g,b^2=eg,d^2=hi,e^2=hi,f^2=i,g^2=hi,b^a=be,c^a=cf,d^a=di,e^a=eh,f^a=fh,d^b=dhi,e^b=ei,f^b=fhi,g^b=gi,e^c=ehi,f^c=fi,g^c=ghi

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 106358);