#703990

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

2072 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)(3,5)(4,6)(7,8)(9,10,13,15)(11,12,16,14)(17,18,22,21)(19,24,20,23)(25,26,28,30)(27,29,32,31);(1,2,5,3)(4,7,8,6)(9,11,13,16)(10,14,15,12)(25,26)(27,31)(28,30)(29,32);(2,3)(6,7)(10,15)(12,14)(17,19,22,20)(18,23,21,24)(25,26,28,30)(27,29,32,31);(1,3,5,2)(4,7,8,6)(9,12)(10,11)(13,14)(15,16)(17,20,22,19)(18,24,21,23)(25,26)(27,29)(28,30)(31,32);(1,4,5,8)(2,6,3,7)(9,11)(10,12)(13,16)(14,15)(17,21,22,18)(19,24,20,23)(25,27)(26,29)(28,32)(30,31)

Polycyclic presentation:

2^9|a^2=gh,b^2=fgi,c^2=fg,d^2=fgh,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, 611186);