#6367439

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

1623 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)(19,26)(20,21)(23,30)(24,28)(27,29)(31,32);(1,2,6,8)(3,11,13,15)(4,7,5,10)(9,14,16,12)(17,18,22,25)(19,23,27,32)(20,28,21,24)(26,30,29,31);(2,8)(7,10)(9,16)(11,15)(17,19)(18,23)(20,26)(21,29)(22,27)(24,31)(25,32)(28,30);(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)(18,24,25,28)(19,26,27,29)(23,31,32,30)

Polycyclic presentation:

2^9|b^2=hi,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	#72339760, #72339782, #72339810
maximal normal subgroup of	#72339760, #72339782, #72339810

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6274635);