#6367670

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
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₄² ▷ C₂² ▷ C₁

1543 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)(3,8)(4,9)(5,7)(6,10)(11,16)(12,14)(13,15)(18,23)(20,21)(24,31)(25,29)(26,32)(27,30);(2,7)(4,14)(5,16)(8,15)(9,13)(10,11)(17,18)(19,26)(20,29)(21,23)(22,25)(24,27)(28,31)(30,32);(1,3)(2,8)(4,11)(5,13)(6,12)(7,15)(9,16)(10,14)(17,18)(19,24)(20,23)(21,29)(22,25)(26,27)(28,32)(30,31);(1,4,6,15)(2,9,10,13)(3,11,12,7)(5,8,16,14)(17,19)(18,24)(20,30)(21,27)(22,28)(23,31)(25,32)(26,29);(2,10)(3,12)(7,11)(9,13)(18,25)(23,29)(24,32)(26,31)

Polycyclic presentation:

2^9|d^2=hi,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.

Relationships

maximal subgroup of	#72283125
maximal normal subgroup of	#72283125

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6274866);