#4328686

order	512
nilpotency class	2
isoclinism	isoclinic class of #4311730
automorphism	Aut(G) order 1048576
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	C₂²×#121	C₂²
LCS	G ▷ C₂⁴ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

2191 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)(3,8)(4,5)(6,7)(9,10)(11,14)(12,13)(15,16)(19,23)(22,24)(27,31)(30,32);(2,7)(3,5)(9,11,15,13)(10,12,16,14)(18,21)(22,24)(26,29)(30,32);(1,3,6,5)(2,8,7,4)(9,12,15,14)(10,11,16,13)(17,18,20,21)(19,22,23,24)(25,26)(27,30)(28,29)(31,32);(1,4,6,8)(2,3,7,5)(9,13,15,11)(10,12,16,14)(25,27)(26,30)(28,31)(29,32);(1,5,6,3)(2,8,7,4)(9,14,15,12)(10,11,16,13)(17,19)(18,22)(20,23)(21,24)(25,27)(26,30)(28,31)(29,32)

Polycyclic presentation:

2^9|b^2=g,c^2=fhi,d^2=f,e^2=f,b^a=bf,c^a=cg,d^a=di,e^a=eh,c^b=ch,e^b=ef,d^c=df

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 4235882);