#6357916

order	512
nilpotency class	3
isoclinism	isoclinic class of #6357891
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	#62541	C₂
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₄×#41 ◁ G
LECS	G ▷ C₄² ▷ C₂² ▷ C₁

1899 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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=fi,c^2=g,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,e^d=eh

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6265112);