#6357451

order	512
nilpotency class	3
isoclinism	isoclinic class of #6357427
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁷×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂²×#127
commutator, Frattini, cosocle	C₄²	C₂⁵
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₄² ▷ C₂² ▷ C₁

1967 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:

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

Polycyclic presentation:

2^9|b^2=f,c^2=g,e^2=i,f^2=h,g^2=i,b^a=bf,c^a=cg,f^a=fh,g^a=gi,d^b=di,e^b=eh,d^c=dh

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72306857, #72306861, #72306928
maximal normal subgroup of	#72306857, #72306861, #72306928

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6264647);