#6471270

order	512
nilpotency class	3
isoclinism	isoclinic class of #6470149
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	#29630	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2296 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1787 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)(19,23)(20,26)(21,25)(22,27)(24,28)(29,32)(30,31);(1,3,5,4)(2,6,7,8)(9,11)(10,13)(12,15)(14,16)(17,19,22,29)(18,23,27,32)(20,30,21,28)(24,26,31,25);(11,15)(13,16)(17,19)(18,24)(20,28)(21,30)(22,29)(23,25)(26,32)(27,31);(1,4,5,3)(2,6,7,8)(11,15)(13,16);(10,14)(13,16)(17,20,22,21)(18,25,27,26)(19,28,29,30)(23,31,32,24)

Polycyclic presentation:

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

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, 6378466);