#6368287

order	512
nilpotency class	3
isoclinism	isoclinic class of #6368046
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

1542 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:

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

Polycyclic presentation:

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

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6275483);