#212010

order	512
nilpotency class	3
isoclinism	isoclinic class of #19552
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁴

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	C₄×#121
prime radical	#19160	C₂
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄×C₈ ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

574 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 4 generators on 32 points:

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

Polycyclic presentation:

2^9|a^2=g,b^2=eh,d^2=g,e^2=h,f^2=i,g^2=hi,b^a=be,c^a=cf,e^a=eh,f^a=fi,c^b=ch,d^b=dhi,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, 119206);