#213343

order	512
nilpotency class	3
isoclinism	isoclinic class of #19733
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	C₂×#127
commutator	C₄²	C₂³×C₄
Frattini, cosocle	C₂×C₄²	C₂⁴
socle	C₂²	C₄×#127
prime radical	#19619	C₂
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄×C₈ ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

679 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:

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

Polycyclic presentation:

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

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, 120539);