#6357015

order	512
nilpotency class	3
isoclinism	isoclinic class of #6356901
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
prime radical	#61995	C₂
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₄×#41 ◁ G
LECS	G ▷ C₄² ▷ C₂² ▷ C₁

1578 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)(3,10,11,16)(5,9,6,15)(8,12,14,13)(17,18)(19,25)(20,26)(21,23)(22,30)(24,27)(28,32)(29,31);(1,3,6,13,7,11,5,12)(2,8,9,10,4,14,15,16)(18,23)(24,31)(25,32)(26,30);(3,11)(8,14)(10,16)(12,13)(17,19)(18,24)(20,27)(21,28)(22,29)(23,31)(25,30)(26,32);(1,4,7,2)(3,8,11,14)(5,9,6,15)(10,12,16,13)(17,19)(18,25)(20,29)(21,28)(22,27)(23,32)(24,30)(26,31);(1,5,7,6)(2,9,4,15)(3,12,11,13)(8,10,14,16)(17,20,21,22)(18,26,23,30)(19,27,28,29)(24,32,31,25)

Polycyclic presentation:

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

Conjugacy classes & character table

65 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6264211);