#212009

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	#3106	C₂²
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄×C₈ ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

557 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=e,d^2=gh,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, 119205);