#6363336

order	512
nilpotency class	3
isoclinism	isoclinic class of #6363286
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	#18810	C₂
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₄² ▷ C₂² ▷ C₁

1549 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)(3,9)(4,14)(5,7)(6,12)(8,15)(10,11)(13,16)(17,18)(19,23)(20,27)(21,29)(22,25)(24,26)(28,31)(30,32);(1,3,5,13)(2,6,7,10)(4,11,15,12)(8,16,14,9)(17,19,22,21)(18,23,25,29)(20,26,28,30)(24,31,32,27);(1,2)(3,10)(4,8)(5,7)(6,13)(9,11)(12,16)(14,15)(17,20,22,28)(18,24,25,32)(19,26,21,30)(23,31,29,27);(3,11)(4,15)(6,16)(8,14)(9,10)(12,13);(2,7)(6,10)(8,14)(9,16)(18,25)(23,29)(24,32)(27,31)

Polycyclic presentation:

2^9|b^2=hi,c^2=i,f^2=h,g^2=i,b^a=bf,c^a=cg,e^a=ehi,f^a=fh,g^a=gi,c^b=ch,d^b=df,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	#72340116, #72340121, #72340134
maximal normal subgroup of	#72340116, #72340121, #72340134

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6270532);