#6357009

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	#61978	C₂
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₄×#41 ◁ G
LECS	G ▷ C₄² ▷ C₂² ▷ C₁

1542 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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=fi,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, 6264205);