#610559

order	512
nilpotency class	2
isoclinism	isoclinic class of #15257
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁵×C₄

Aliases

No aliases for this group.

Subgroups

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

2024 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)(3,6)(4,5)(7,8)(9,10)(11,14)(12,13)(15,16)(18,21)(22,23)(25,26)(27,29)(28,30)(31,32);(2,5)(3,7)(17,18,20,21)(19,23,24,22)(25,27,28,31)(26,29,30,32);(10,13)(14,16)(17,19,20,24)(18,22,21,23);(1,3,4,7)(2,6,5,8)(18,21)(22,23)(26,30)(29,32);(9,11,12,15)(10,14,13,16)

Polycyclic presentation:

2^9|b^2=hi,c^2=h,d^2=fh,e^2=g,b^a=bf,c^a=cg,d^a=di,c^b=ch,d^b=df

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, 517755);