#198367

order	512
nilpotency class	3
isoclinism	isoclinic class of #198171
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	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	C₂×#532	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

1065 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=egi,c^2=i,d^2=h,e^2=hi,f^2=hi,g^2=i,b^a=be,c^a=cf,d^a=di,e^a=eh,d^b=di,e^b=ei,f^b=fhi,g^b=gi,d^c=dh,e^c=ehi,f^c=fhi,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, 105563);