#189655

order	512
nilpotency class	3
isoclinism	isoclinic class of #19328
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	C₂×#121
commutator	C₄²	C₂³×C₄
Frattini, cosocle	C₂×C₄²	C₂⁴
socle	C₂²	C₂×#380
prime radical	#3444	C₂²
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

868 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 96851);