#7198644

order	512
nilpotency class	3
isoclinism	isoclinic class of #7021351
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁸

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×D₄²
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
prime radical	#2011	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1624 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 64

A minimal faithful permutation representation — 5 generators on 32 points:

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

Polycyclic presentation:

2^9|a^2=fhi,b^2=h,c^2=gh,d^2=f,e^2=hi,f^2=i,g^2=i,b^a=bf,c^a=cg,e^a=eh,g^a=gi,c^b=ch,d^b=df,e^b=eh,f^b=fi,d^c=dh

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7105840);