#6357786

order	512
nilpotency class	3
isoclinism	isoclinic class of #6357777
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
prime radical	#60700	C₂
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₄² ▷ C₂² ▷ C₁

1977 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)(3,10)(4,8)(5,14)(6,9)(7,13)(11,15)(12,16)(17,18,21,24)(19,25,26,32)(20,28,29,23)(22,30,31,27);(1,3,5,13,6,12,4,11)(2,7,8,10,9,15,14,16);(17,19,20,27,21,26,29,30)(18,22,23,25,24,31,28,32);(1,4,6,5)(2,8,9,14)(3,11,12,13)(7,10,15,16)(19,26)(22,31)(25,32)(27,30);(3,12)(7,15)(10,16)(11,13)

Polycyclic presentation:

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

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72224737, #72306706, #72306707, …
maximal normal subgroup of	#72224737, #72306706, #72306707, …
maximal quotient of	#72224725

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6264982);