#6360367

order	512
nilpotency class	3
isoclinism	isoclinic class of #6360176
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
prime radical	#61433	C₂
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₄² ▷ C₂² ▷ C₁

1537 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 5 generators on 32 points:

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

Polycyclic presentation:

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

Conjugacy classes & character table

65 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6267563);