#741860

order	512
nilpotency class	2
isoclinism	isoclinic class of #686200
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

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

1734 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:

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

Polycyclic presentation:

2^9|a^2=gi,b^2=gi,c^2=fgi,d^2=fghi,e^2=gh,b^a=bf,c^a=cg,d^a=di,c^b=ch,e^b=ef,d^c=df

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 649056);