#6362180

order	512
nilpotency class	3
isoclinism	isoclinic class of #6362118
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

1569 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)(3,7)(4,11)(5,13)(6,9)(8,10)(12,15)(14,16)(18,23)(20,21)(24,31)(25,29)(26,32)(27,30);(1,3,6,5)(2,7,9,13)(4,10,12,14)(8,15,16,11)(17,18)(19,26)(20,29)(21,23)(22,25)(24,27)(28,31)(30,32);(1,4,6,12)(2,8,9,16)(3,10,5,14)(7,15,13,11)(17,18)(19,24)(20,23)(21,29)(22,25)(26,27)(28,32)(30,31);(17,19,22,28)(18,24,25,32)(20,30,21,27)(23,31,29,26);(2,9)(7,13)(8,16)(11,15)

Polycyclic presentation:

2^9|b^2=hi,c^2=hi,d^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,f^b=fh,g^c=gi,f^d=fh

Conjugacy classes & character table

65 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72283748, #72283889
maximal normal subgroup of	#72283748, #72283889

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6269376);