#6719191

order	512
nilpotency class	3
isoclinism	isoclinic class of #6667604
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	#28467	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2399 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1563 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 256

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

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

Polycyclic presentation:

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

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72257411, #72257477, #72264159
maximal normal subgroup of	#72257411, #72257477, #72264159
maximal quotient of	#72264526

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6626387);