#6369140

order	512
nilpotency class	3
isoclinism	isoclinic class of #6369070
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

1616 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72306665, #72306668, #72306686
maximal normal subgroup of	#72306665, #72306668, #72306686

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6276336);