#559909

order	512
nilpotency class	2
isoclinism	isoclinic class of #45438
automorphism	?
homology	Abelianization (H₁) = C₂⁴×C₄ · Schur multiplier (H₂) = C₂⁸

Aliases

No aliases for this group.

Subgroups

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

2366 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 512

A minimal faithful permutation representation — 5 generators on 32 points:

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

Polycyclic presentation:

2^9|a^2=f,b^2=i,c^2=h,d^2=f,e^2=g,b^a=bf,c^a=cg,d^a=df,e^a=eg,c^b=ch,d^b=dfg,e^b=ef

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, 467105);