#6364607

order	512
nilpotency class	3
isoclinism	isoclinic class of #6364250
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

1580 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

65 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72306797, #72306850, #72306858, …
maximal normal subgroup of	#72306797, #72306850, #72306858, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6271803);