#6356560

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

1613 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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=fh,c^2=i,d^2=h,e^2=h,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.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6263756);