#6476549

order	512
nilpotency class	3
isoclinism	isoclinic class of #6470395
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	C₂²×#121	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×#117 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1751 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)(9,10)(11,13)(12,14)(15,16)(18,24)(19,29)(21,22)(25,31)(26,30)(27,28);(9,11)(10,13)(12,15)(14,16)(17,18,22,24,23,28,21,27)(19,30,20,31,29,25,32,26);(2,5)(6,8)(11,15)(13,16)(17,19,23,29)(18,25,28,30)(20,21,32,22)(24,26,27,31);(11,15)(13,16)(17,20)(18,26)(19,21)(22,29)(23,32)(24,30)(25,27)(28,31);(1,3,4,7)(2,6,5,8)(10,14)(13,16)(17,21,23,22)(18,27,28,24)(19,20,29,32)(25,26,30,31)

Polycyclic presentation:

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

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, 6383745);