#1793073

order	512
nilpotency class	2
isoclinism	isoclinic class of #1506685
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁷

Aliases

No aliases for this group.

Subgroups

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

2554 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)(9,10,13,15)(11,12,16,14)(17,18)(19,23)(20,21)(22,24)(25,26,28,30)(27,31,32,29);(1,2,4,6)(3,5,7,8)(9,11,13,16)(10,14,15,12)(18,21)(19,24)(25,26,28,30)(27,29,32,31);(3,7)(5,8)(10,15)(12,14)(17,18,20,21)(19,22,24,23)(25,26)(27,29)(28,30)(31,32);(1,3,4,7)(2,5,6,8)(9,12,13,14)(10,11,15,16)(17,18)(19,22)(20,21)(23,24)(25,26)(27,29)(28,30)(31,32);(1,2)(3,5)(4,6)(7,8)(9,11)(10,14)(12,15)(13,16)(17,19,20,24)(18,22,21,23)(25,27,28,32)(26,29,30,31)

Polycyclic presentation:

2^9|a^2=i,b^2=fh,c^2=gi,d^2=fgi,e^2=h,b^a=bf,c^a=cg,d^a=di,e^a=eg,c^b=ch,d^b=dg,e^b=ef,d^c=df

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 1700269);