#6367406

order	512
nilpotency class	3
isoclinism	isoclinic class of #6367198
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

1669 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)(8,10)(9,15)(11,16)(12,14)(17,18)(19,23)(20,26)(21,25)(22,27)(24,28)(29,32)(30,31);(1,2)(3,11)(4,7)(5,10)(6,8)(9,12)(13,15)(14,16)(17,19)(18,23)(20,30)(21,28)(22,29)(24,25)(26,31)(27,32);(2,8)(7,10)(9,16)(11,15)(17,19)(18,24)(20,28)(21,30)(22,29)(23,25)(26,32)(27,31);(1,3,6,13)(2,9,8,16)(4,12,5,14)(7,15,10,11);(1,4,6,5)(2,10,8,7)(3,12,13,14)(9,11,16,15)(17,20,22,21)(18,25,27,26)(19,28,29,30)(23,31,32,24)

Polycyclic presentation:

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

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72339736, #72339751, #72339810
maximal normal subgroup of	#72339736, #72339751, #72339810

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6274602);