#6357797

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
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₄² ▷ C₂² ▷ C₁

2004 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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=fh,c^2=g,d^2=h,e^2=i,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	#72231251, #72231256, #72231262, …
maximal normal subgroup of	#72231251, #72231256, #72231262, …
maximal quotient of	#72280153, C₂×#6357797, #72309562

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6264993);