#198319

order	512
nilpotency class	3
isoclinism	isoclinic class of #198144
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	C₂⁴	C₄×D₄
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

1062 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=egh,c^2=i,d^2=h,e^2=hi,f^2=hi,g^2=i,b^a=be,c^a=cf,e^a=eh,d^b=di,e^b=ei,f^b=fhi,g^b=gi,e^c=ehi,f^c=fhi,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, 105515);