#6412327

order	512
nilpotency class	2
isoclinism	isoclinic class of #37436
automorphism	?
homology	Abelianization (H₁) = C₂⁴×C₄ · Schur multiplier (H₂) = C₂⁶×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, Frattini, cosocle	C₂²×C₄	C₂⁵
commutator, socle	C₂³	C₂⁴×C₄
prime radical	#34609	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂²×C₄ ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1738 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)(3,5)(4,6)(7,8)(10,14)(11,12)(17,18,22,26,21,23,20,25)(19,24,29,32,28,30,27,31);(1,3)(2,5)(4,7)(6,8)(9,10)(11,16)(12,15)(13,14)(18,23)(24,30)(25,26)(31,32);(2,6)(3,7)(9,11,13,12)(10,15,14,16)(17,19,20,27,21,28,22,29)(18,24,25,31,23,30,26,32);(2,6)(5,8)(9,11,13,12)(10,15,14,16)(17,20,21,22)(18,25,23,26)(19,27,28,29)(24,31,30,32);(9,12,13,11)(10,15,14,16)(17,20,21,22)(18,25,23,26)(19,27,28,29)(24,31,30,32)

Polycyclic presentation:

2^9|a^2=h,c^2=fh,d^2=f,e^2=f,h^2=i,b^a=bf,c^a=cg,d^a=dg,e^a=efi,c^b=cg,d^b=dfi

Conjugacy classes & character table

104 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6319523);