#615872

order	512
nilpotency class	2
isoclinism	isoclinic class of #612424
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

2002 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:

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

Polycyclic presentation:

2^9|a^2=f,b^2=f,c^2=i,d^2=fh,e^2=g,b^a=bf,c^a=cg,d^a=di,e^a=ef,c^b=ch

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, 523068);