#6614747

order	512
nilpotency class	3
isoclinism	isoclinic class of #6574015
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 ◁ #2290 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1642 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)(4,7)(5,6)(9,10)(11,14)(12,15)(13,16)(17,18,22,25)(19,26,29,32)(20,24,21,30)(23,27,31,28);(2,6)(7,8)(9,11)(10,13)(12,16)(14,15)(17,19,20,27,22,29,21,28)(18,23,30,32,25,31,24,26);(1,3,5,4)(2,7,6,8)(11,16)(13,14)(17,18)(19,23)(20,30)(21,24)(22,25)(26,28)(27,32)(29,31);(1,4,5,3)(2,7,6,8)(9,11)(10,14)(12,16)(13,15)(17,20,22,21)(18,24,25,30)(19,27,29,28)(23,26,31,32);(11,16)(13,14)

Polycyclic presentation:

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

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, 6521943);