#6431335

order	512
nilpotency class	2
isoclinism	isoclinic class of #37644
automorphism	?
homology	Abelianization (H₁) = C₂⁴×C₄ · Schur multiplier (H₂) = C₂⁷

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, Frattini, cosocle	C₂²×C₄	C₂⁵
commutator, socle	C₂³	C₂⁴×C₄
prime radical	#2676	C₄
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂²×C₄ ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1433 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)(3,8)(4,7)(5,6)(9,10)(11,15)(12,13)(14,16)(17,18,23,26,22,24,20,31)(19,28,21,32,30,25,29,27);(2,6)(3,4)(9,10)(11,14)(12,13)(15,16)(18,24)(25,28)(26,31)(27,32);(1,3,5,4)(2,7,6,8)(10,13)(14,15)(17,19,20,29,22,30,23,21)(18,25,31,32,24,28,26,27);(1,4,5,3)(2,8,6,7)(9,11,12,16)(10,14,13,15)(17,20,22,23)(18,26,24,31)(19,29,30,21)(25,27,28,32);(2,6)(7,8)(17,21,23,30,22,29,20,19)(18,27,26,28,24,32,31,25)

Polycyclic presentation:

2^9|a^2=h,c^2=fh,d^2=g,e^2=h,h^2=i,b^a=bf,c^a=cg,d^a=dg,e^a=ef,c^b=cgi,d^b=dfi

Conjugacy classes & character table

116 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6338531);