#6364166

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

1613 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)(3,8)(4,9)(5,7)(6,10)(11,15)(12,16)(13,14)(17,18,22,26)(19,23,28,32)(20,25,21,30)(24,29,31,27);(2,7)(3,5)(4,11)(8,10)(9,16)(12,13)(17,19)(18,23)(20,29)(21,27)(22,28)(24,30)(25,31)(26,32);(1,3,6,5)(2,8,10,7)(4,11,12,13)(9,15,16,14)(17,19)(18,24)(20,27)(21,29)(22,28)(23,30)(25,32)(26,31);(1,4,6,12)(2,9,10,16)(3,11,5,13)(7,14,8,15);(2,10)(7,8)(9,16)(14,15)(17,20,22,21)(18,25,26,30)(19,27,28,29)(23,24,32,31)

Polycyclic presentation:

2^9|a^2=i,c^2=h,d^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,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	#72339713, #72339719, #72339810
maximal normal subgroup of	#72339713, #72339719, #72339810

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6271362);