#452526

order	512
nilpotency class	3
isoclinism	isoclinic class of #452526
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂	#16051
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂×#535	C₂²
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

915 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

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

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

Polycyclic presentation:

2^9|a^2=egh,b^2=f,c^2=ef,d^2=ef,f^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,f^a=fi,c^b=cg,d^b=de,e^b=ei,h^b=hi,g^c=gi,h^c=hi,e^d=ei,f^d=fi

Conjugacy classes & character table

59 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 359722);