#189653

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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=e,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, 96849);