#6366677

order	512
nilpotency class	3
isoclinism	isoclinic class of #6366542
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

1570 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:

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

Polycyclic presentation:

2^9|a^2=hi,d^2=h,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,e^c=eh,g^c=gi,f^d=fh

Conjugacy classes & character table

65 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72284090
maximal normal subgroup of	#72284090

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6273873);