#560402

order	512
nilpotency class	2
isoclinism	isoclinic class of #47826
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

2214 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:

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

Polycyclic presentation:

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

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, 467598);