#732073

order	512
nilpotency class	2
isoclinism	isoclinic class of #686200
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, commutator, Frattini, socle, cosocle	C₂⁴	C₂⁵
prime radical	C₂³×D₄	C₂³
LCS	G ▷ C₂⁴ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

1986 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:

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

Polycyclic presentation:

2^9|a^2=fgi,b^2=gi,c^2=g,d^2=ghi,e^2=fgh,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, 639269);