#6690800

order	512
nilpotency class	3
isoclinism	isoclinic class of #6667580
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
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2397 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2214 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72282572, #72282621, #72284374
maximal normal subgroup of	#72282572, #72282621, #72284374

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6597996);