#6362709

order	512
nilpotency class	3
isoclinism	isoclinic class of #6362510
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁸

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×D₄²
commutator, Frattini, cosocle	C₄²	C₂⁵
prime radical	#61432	C₂
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₄² ▷ C₂² ▷ C₁

1550 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

71 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6269905);