#6412322

order	512
nilpotency class	2
isoclinism	isoclinic class of #40484
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	#34977	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂²×C₄ ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1765 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)(17,18,22,25,21,23,20,30)(19,26,27,31,29,24,28,32);(1,3)(2,5)(4,7)(6,8)(9,10)(11,16)(12,15)(13,14)(18,23)(24,26)(25,30)(31,32);(2,6)(3,7)(9,11,13,12)(10,15,14,16)(17,19,20,28,21,29,22,27)(18,24,30,31,23,26,25,32);(2,6)(5,8)(9,11,13,12)(10,15,14,16)(17,20,21,22)(18,25,23,30)(19,27,29,28)(24,31,26,32);(9,12,13,11)(10,15,14,16)(17,20,21,22)(18,25,23,30)(19,28,29,27)(24,32,26,31)

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=ef,c^b=cgi,d^b=dfi,d^c=di

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, 6319518);