#6366148

order	512
nilpotency class	3
isoclinism	isoclinic class of #6365694
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

1609 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)(3,13)(4,12)(5,10)(6,9)(7,11)(8,14)(15,16)(17,18)(19,24)(20,29)(21,27)(22,31)(23,26)(25,30)(28,32);(1,3,7,15)(2,8,11,5)(4,14,6,10)(9,13,12,16)(17,19,23,20)(18,24,26,29)(21,30,22,28)(25,31,32,27);(1,4,7,6)(2,9,11,12)(3,14,15,10)(5,16,8,13)(17,20,23,19)(18,25,26,32)(21,30,22,28)(24,31,29,27);(1,5)(2,10)(3,9)(4,16)(6,13)(7,8)(11,14)(12,15)(18,26)(24,29)(25,32)(27,31);(2,11)(3,15)(9,12)(10,14)(17,21,23,22)(18,27,26,31)(19,28,20,30)(24,32,29,25)

Polycyclic presentation:

2^9|b^2=hi,c^2=hi,e^2=i,f^2=h,g^2=i,b^a=bf,c^a=cg,d^a=di,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

59 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72339696, #72339705, #72339736
maximal normal subgroup of	#72339696, #72339705, #72339736

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6273344);