#6474150

order	512
nilpotency class	3
isoclinism	isoclinic class of #6470922
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	C₂×#469	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2296 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1391 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 5 generators on 32 points:

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

Polycyclic presentation:

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

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6381346);