#6365683

order	512
nilpotency class	3
isoclinism	isoclinic class of #6365522
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	C₄²	C₂⁵
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₄² ▷ C₂² ▷ C₁

1606 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 128

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

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

Polycyclic presentation:

2^9|a^2=hi,b^2=h,c^2=h,d^2=h,e^2=hi,f^2=h,g^2=i,b^a=bf,c^a=cg,d^a=di,e^a=eh,f^a=fh,g^a=gi,c^b=ch,d^b=df,e^b=ei,f^b=fh,g^c=gi,f^d=fh

Conjugacy classes & character table

56 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72306907, #72306967, #72306971
maximal normal subgroup of	#72306907, #72306967, #72306971

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6272879);