#640967

order	512
nilpotency class	2
isoclinism	isoclinic class of #635984
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
prime radical	C₂²×#121	C₂²
LCS	G ▷ C₂⁴ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

2432 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)(3,8)(4,6)(5,7)(9,10,13,14)(11,15,16,12)(17,18,20,22)(19,23,24,21)(25,26,29,32)(27,31,28,30);(1,3,6,5)(2,7,4,8)(9,11)(10,12)(13,16)(14,15)(17,18)(19,21)(20,22)(23,24)(25,27,29,28)(26,30,32,31);(2,4)(7,8)(11,16)(12,15)(17,18,20,22)(19,21,24,23)(25,26,29,32)(27,30,28,31);(1,4,6,2)(3,8,5,7)(9,12)(10,11)(13,15)(14,16)(17,18)(19,21)(20,22)(23,24)(25,26)(27,30)(28,31)(29,32);(1,5,6,3)(2,8,4,7)(9,10)(11,12)(13,14)(15,16)(17,19)(18,21)(20,24)(22,23)(25,28,29,27)(26,31,32,30)

Polycyclic presentation:

2^9|a^2=fg,b^2=fh,c^2=gi,d^2=g,e^2=fh,b^a=bf,c^a=cg,d^a=di,e^a=ef,c^b=ch,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, 548163);