#6587789

order	512
nilpotency class	3
isoclinism	isoclinic class of #6573485
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁷

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂³×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
prime radical	#25124	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄×#123 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1760 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)(3,7)(4,8)(5,6)(9,10)(11,14)(12,13)(15,16)(17,18)(19,27)(20,25)(21,24)(22,26)(23,30)(28,31)(29,32);(2,6)(7,8)(17,19)(18,23)(20,29)(21,30)(22,28)(24,27)(25,26)(31,32);(1,3,5,4)(2,7,6,8)(10,13)(14,16)(18,24)(19,28)(20,30)(22,29)(23,31)(25,27);(9,11,12,15)(10,14,13,16)(17,20,21,28)(18,25,24,31)(19,29,30,22)(23,26,27,32);(1,4,5,3)(2,8,6,7)(18,24)(19,28)(20,30)(22,29)(23,31)(25,27)

Polycyclic presentation:

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

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72282526, #72283180, #72283382
maximal normal subgroup of	#72282526, #72283180, #72283382

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6494985);