#6451785

order	512
nilpotency class	3
isoclinism	isoclinic class of #6451783
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂¹⁰

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂	C₂×#2404
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

3903 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)(3,13)(4,9)(5,10)(6,11)(7,12)(8,16)(14,29)(15,23)(17,24)(18,32)(19,26)(20,28)(21,27)(22,31)(25,30);(1,3)(2,8)(4,18)(5,15)(6,16)(7,17)(9,25)(10,13)(11,24)(12,23)(14,20)(19,30)(21,31)(22,26)(27,29)(28,32);(1,2)(3,13,17,24)(4,9)(5,11)(6,10)(7,12)(8,16,23,15)(14,29,30,25)(18,22,31,32)(19,28)(20,26)(21,27);(1,4)(2,9)(3,14,17,30)(5,19)(6,20)(7,21)(8,22,23,32)(10,26)(11,28)(12,27)(13,29,24,25)(15,18,16,31);(1,5,7,6)(2,10,12,11)(3,15,17,16)(4,19,21,20)(8,13,23,24)(9,26,27,28)(14,18,30,31)(22,29,32,25)

Polycyclic presentation:

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

Conjugacy classes & character table

77 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72251414, #72251420, #72251427, …
maximal normal subgroup of	#72251414, #72251420, #72251427, …
maximal quotient of	#72255859, #72318695

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6358981);