#713006

order	512
nilpotency class	2
isoclinism	isoclinic class of #686200
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁸

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

2986 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:

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

Polycyclic presentation:

2^9|a^2=f,b^2=fgi,c^2=hi,d^2=g,e^2=f,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, 620202);