#6358039

order	512
nilpotency class	3
isoclinism	isoclinic class of #6358021
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁷×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂²×#127
commutator, Frattini, cosocle	C₄²	C₂⁵
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₄×#41 ◁ G
LECS	G ▷ C₄² ▷ C₂² ▷ C₁

2086 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)(3,10)(4,9)(5,14)(6,8)(7,12)(11,15)(13,16)(19,25)(20,27)(22,28)(24,30)(26,31)(29,32);(1,3,4,11,6,13,5,12)(2,7,14,16,8,15,9,10)(17,18)(19,22)(20,24)(21,23)(25,28)(26,29)(27,30)(31,32);(2,8)(7,15)(9,14)(10,16)(17,19,20,25,21,26,27,31)(18,22,24,28,23,29,30,32);(1,2)(3,7)(4,14)(5,9)(6,8)(10,12)(11,16)(13,15)(19,26)(22,29)(25,31)(28,32);(1,4,6,5)(2,9,8,14)(3,11,13,12)(7,10,15,16)(18,23)(22,29)(24,30)(28,32)

Polycyclic presentation:

2^9|b^2=fh,c^2=gh,e^2=h,f^2=h,g^2=i,b^a=bf,c^a=cg,f^a=fh,g^a=gi,e^b=ei,d^c=dhi,e^d=eh

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72339709, #72339799, #72339835, …
maximal normal subgroup of	#72339709, #72339799, #72339835, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6265235);