#6347366

order	512
nilpotency class	3
isoclinism	isoclinic class of #6347363
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁹

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂²×#120
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
prime radical	#62818	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂⁴×C₄ ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₁

3228 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

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

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

Polycyclic presentation:

2^9|a^2=h,d^2=h,f^2=h,b^a=bf,c^a=cg,d^a=di,f^a=fh,e^b=eh,f^b=fh,g^b=gi,d^c=dh,f^c=fi

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72232857, #72234669, #72234673, …
maximal normal subgroup of	#72232857, #72234669, #72234673, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6254562);