#6578365

order	512
nilpotency class	3
isoclinism	isoclinic class of #6573549
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁸

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂³×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄²×D₄ ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2446 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 20
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 5 generators on 20 points:

(5,6,9,10)(7,11,8,12)(14,17)(15,18)(19,20);(1,2)(3,4)(6,10)(11,12)(13,14,16,17)(15,18,19,20);(2,4)(13,15,16,19)(14,18,17,20);(1,2)(3,4)(5,7,9,8)(6,11,10,12)(13,14,16,17)(15,18,19,20);(2,4)(5,8,9,7)(6,12,10,11)

Polycyclic presentation:

2^9|a^2=fi,b^2=i,c^2=i,d^2=f,e^2=fi,g^2=i,b^a=bf,c^a=cg,d^a=di,g^a=gi,c^b=ch,e^b=eh,d^c=dh,g^c=gi,e^d=eh

Conjugacy classes & character table

83 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72340619, #72340639, #72340643
maximal normal subgroup of	#72340619, #72340639, #72340643

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6485561);