#6364139

order	512
nilpotency class	3
isoclinism	isoclinic class of #6363930
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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72306900, #72306963, #72306967, …
maximal normal subgroup of	#72306900, #72306963, #72306967, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6271335);