#6368495

order	512
nilpotency class	3
isoclinism	isoclinic class of #6368046
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

1610 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72339693, #72339709, #72339760
maximal normal subgroup of	#72339693, #72339709, #72339760

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6275691);