#6367607

order	512
nilpotency class	3
isoclinism	isoclinic class of #6367534
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	#3546	C₂²
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₄² ▷ C₂² ▷ C₁

1513 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)(3,12,13,8)(4,9,15,14)(5,7,16,11)(17,18)(19,25)(20,28)(21,23)(22,26)(24,30)(27,32)(29,31);(2,7)(4,12)(5,16)(8,15)(9,14)(10,11)(17,18)(19,22)(20,24)(21,23)(25,26)(27,31)(28,30)(29,32);(1,3)(2,8)(4,11)(5,14)(6,13)(7,15)(9,16)(10,12)(17,19,21,27)(18,22,23,31)(20,29,30,26)(24,32,28,25);(1,4,6,15)(2,9,10,14)(3,11,13,7)(5,8,16,12);(2,10)(3,13)(7,11)(9,14)(18,23)(22,31)(24,28)(25,32)

Polycyclic presentation:

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

Conjugacy classes & character table

59 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6274803);