#6690437

order	512
nilpotency class	3
isoclinism	isoclinic class of #6667460
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 ◁ C₄×#121 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2054 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)(4,6)(5,8)(9,10)(11,14)(12,16)(13,15)(17,18,21,24)(19,22,27,31)(20,23,28,32)(25,29,26,30);(1,3,4,8)(2,5,6,7)(9,11,13,12)(10,14,15,16)(17,19)(18,22)(20,25)(21,27)(23,29)(24,31)(26,28)(30,32);(9,12,13,11)(10,14,15,16)(19,25)(20,28)(22,29)(23,32)(26,27)(30,31);(2,6)(5,7);(3,8)(5,7)(19,25)(20,28)(22,29)(23,32)(26,27)(30,31)

Polycyclic presentation:

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

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72282558, #72282817, #72282987
maximal normal subgroup of	#72282558, #72282817, #72282987

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6597633);