#6364268

order	512
nilpotency class	3
isoclinism	isoclinic class of #6364250
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

1724 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

65 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72306948, #72307008, #72307043
maximal normal subgroup of	#72306948, #72307008, #72307043

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6271464);