#6357792

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

2088 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)(4,9)(5,14)(6,8)(7,11)(12,16)(13,15)(17,18)(19,25)(20,22)(21,29)(23,27)(24,28)(26,31)(30,32);(1,3,5,11,6,12,4,13)(2,7,9,10,8,15,14,16)(18,22)(23,30)(24,29)(25,31);(2,8)(7,15)(9,14)(10,16)(17,19,21,27,20,26,28,32)(18,23,24,25,22,30,29,31);(19,26)(23,30)(25,31)(27,32);(1,4,6,5)(2,9,8,14)(3,11,12,13)(7,16,15,10)

Polycyclic presentation:

2^9|b^2=fi,c^2=gh,e^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	#72306696, #72306743, #72306765
maximal normal subgroup of	#72306696, #72306743, #72306765

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6264988);