#713009

order	512
nilpotency class	2
isoclinism	isoclinic class of #686200
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁷×C₄

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

2932 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)(11,15)(13,16)(17,18)(19,24)(20,21)(22,23)(25,30)(26,31)(27,32)(28,29);(6,8)(9,10,12,14)(11,13,15,16)(18,23)(19,26)(21,25)(27,29);(1,2,4,3)(5,6,7,8)(9,10)(11,13)(12,14)(15,16)(17,19,22,26)(18,24,23,31)(20,29,30,27)(21,28,25,32);(1,3,4,2)(9,11)(10,13)(12,15)(14,16)(17,20)(18,25)(19,27)(21,23)(22,30)(24,28)(26,29)(31,32);(17,21,22,25)(18,20,23,30)(19,28,26,32)(24,29,31,27)

Polycyclic presentation:

2^9|b^2=fgi,c^2=fhi,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, 620205);