#6356598

order	512
nilpotency class	3
isoclinism	isoclinic class of #6356533
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁸

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂²×#121
commutator, Frattini, cosocle	C₄²	C₂⁵
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₄×#41 ◁ G
LECS	G ▷ C₄² ▷ C₂² ▷ C₁

1649 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)(18,22)(19,26)(20,23)(25,30)(27,28)(29,31);(1,2,4,6,5,8,11,13)(3,7,10,12,9,14,15,16)(17,18,23,30,21,25,20,22)(19,24,26,31,28,32,27,29);(17,19,21,28)(18,24,25,32)(20,27,23,26)(22,29,30,31);(1,3)(2,7)(4,10)(5,9)(6,12)(8,14)(11,15)(13,16);(3,9)(7,14)(10,15)(12,16)

Polycyclic presentation:

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

Conjugacy classes & character table

77 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72283494, #72284038
maximal normal subgroup of	#72283494, #72284038

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6263794);