#191313

order	512
nilpotency class	3
isoclinism	isoclinic class of #191312
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×#383
commutator	C₄²	C₂³×C₄
Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	C₂⁴	#126
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

456 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)(3,11,13,16)(4,8,14,7)(5,9)(6,10)(12,15)(17,18)(19,27,29,23)(20,25)(21,24)(22,26)(28,32,30,31);(1,3,12,14,6,13,5,4)(2,7,9,11,10,8,15,16)(17,19,21,28,22,29,20,30)(18,23,24,31,26,27,25,32);(1,4,5,13,6,14,12,3)(2,8,9,16,10,7,15,11)(18,24,26,25)(23,31,27,32);(17,20,22,21)(18,25,26,24)(19,28,29,30)(23,31,27,32)

Polycyclic presentation:

2^9|a^2=g,b^2=ef,c^2=e,d^2=h,e^2=hi,f^2=h,b^a=be,c^a=cf,e^a=eh,f^a=fi,d^b=dh,g^b=gi,g^c=ghi

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, 98509);