#6506995

order	512
nilpotency class	3
isoclinism	isoclinic class of #6500385
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁷

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂³×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
prime radical	#2408	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄²×D₄ ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1845 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)(3,6)(4,5)(7,8)(17,18)(19,25)(20,22)(21,29)(23,27)(24,28)(26,31)(30,32);(2,5)(6,8)(9,10)(11,15)(12,14)(13,16)(18,22)(23,30)(24,29)(25,31);(9,11)(10,13)(12,16)(14,15)(17,19,21,27,20,26,28,32)(18,23,24,25,22,30,29,31);(1,3,4,7)(2,6,5,8)(11,16)(13,15)(19,26)(23,30)(25,31)(27,32);(10,14)(13,15)

Polycyclic presentation:

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

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6414191);