#6741656

order	512
nilpotency class	3
isoclinism	isoclinic class of #6667836
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	#2537	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄×#123 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1546 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)(11,15)(12,16)(13,14)(17,18,21,23)(19,27,29,24)(20,31,28,25)(22,26,30,32);(1,3)(2,5)(4,8)(6,7)(18,23)(19,28)(20,29)(22,30)(24,25)(27,31);(2,6)(5,7)(10,14)(15,16)(17,19,21,29)(18,24,23,27)(20,30,28,22)(25,32,31,26);(2,6)(5,7)(9,11,13,12)(10,15,14,16)(18,23)(19,29)(20,28)(26,32);(3,8)(5,7)(9,12,13,11)(10,16,14,15)(17,20)(18,25)(19,30)(21,28)(22,29)(23,31)(24,32)(26,27)

Polycyclic presentation:

2^9|a^2=i,c^2=i,d^2=fg,e^2=fg,h^2=i,b^a=bf,c^a=cg,d^a=df,e^a=ei,c^b=ch,e^b=efh,h^b=hi,d^c=di,h^c=hi,e^d=ei,h^e=hi

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6648852);