#6552486

order	512
nilpotency class	3
isoclinism	isoclinic class of #6533105
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
prime radical	C₂×#3309	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄²×D₄ ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1963 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 28
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 5 generators on 28 points:

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

Polycyclic presentation:

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

Conjugacy classes & character table

77 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6459682);