#6476905

order	512
nilpotency class	3
isoclinism	isoclinic class of #6470003
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	#2847	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×#117 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1826 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)(3,8)(4,6)(5,7)(9,10)(11,13)(12,14)(15,16)(17,18,22,26,23,27,21,30)(19,25,28,32,29,31,20,24);(9,11)(10,13)(12,15)(14,16)(17,19,23,29)(18,24,27,32)(20,21,28,22)(25,26,31,30);(1,3,5,4)(2,6,7,8)(11,15)(13,16)(17,20)(18,24)(19,22)(21,29)(23,28)(25,26)(27,32)(30,31);(11,15)(13,16)(17,19)(18,25)(20,21)(22,28)(23,29)(24,30)(26,32)(27,31);(1,4,5,3)(2,6,7,8)(10,14)(13,16)(17,21,23,22)(18,26,27,30)(19,28,29,20)(24,31,32,25)

Polycyclic presentation:

2^9|a^2=f,b^2=i,c^2=g,e^2=gi,f^2=i,b^a=bf,c^a=cg,e^a=eh,c^b=ch,d^b=dh,f^b=fi,d^c=di,e^c=ei,e^d=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, 6384101);