#6369197

order	512
nilpotency class	3
isoclinism	isoclinic class of #6369150
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

1618 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)(9,16)(10,14)(11,13)(19,26)(20,21)(23,31)(24,29)(27,32)(28,30);(1,2)(3,8)(4,7)(5,14)(6,10)(9,11)(12,16)(13,15)(17,18)(19,27)(20,29)(21,24)(22,25)(23,28)(26,32)(30,31);(1,3)(2,8)(4,13)(5,11)(6,12)(7,15)(9,14)(10,16)(17,19)(18,23)(20,26)(21,30)(22,28)(24,32)(25,27)(29,31);(1,3)(2,9)(4,11)(5,13)(6,12)(7,16)(8,14)(10,15)(17,18)(19,23)(20,29)(21,24)(22,25)(26,31)(27,28)(30,32);(1,4,6,5)(2,7,10,14)(3,11,12,13)(8,9,16,15)(17,20,22,21)(18,24,25,29)(19,26,28,30)(23,32,27,31)

Polycyclic presentation:

2^9|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=ci,d^b=df,e^b=ei,f^b=fh,e^c=eh,g^c=gi,e^d=ei,f^d=fh

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72306678, #72306684, #72306689
maximal normal subgroup of	#72306678, #72306684, #72306689

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6276393);