#6356822

order	512
nilpotency class	3
isoclinism	isoclinic class of #6356733
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

1624 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)(4,12)(7,13)(8,14)(9,16)(11,15)(17,18)(19,21)(20,28)(22,27)(23,25)(24,29)(26,31)(30,32);(1,2,4,6,5,8,12,14)(3,9,11,16,10,7,15,13)(17,19,21,18,23,27,22,25)(20,26,29,24,31,28,30,32);(17,20)(18,24)(19,26)(21,29)(22,30)(23,31)(25,32)(27,28);(1,3,5,10)(2,7,8,9)(4,11,12,15)(6,13,14,16)(17,21,23,22)(18,19,25,27)(20,29,31,30)(24,26,32,28);(3,10)(7,9)(11,15)(13,16)

Polycyclic presentation:

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

Conjugacy classes & character table

71 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72338420, #72338422, #72338448
maximal normal subgroup of	#72338420, #72338422, #72338448

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6264018);