#4329116

order	512
nilpotency class	2
isoclinism	isoclinic class of #4311730
automorphism	Aut(G) order 1048576
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁵×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

2216 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)(3,7)(4,8)(5,6)(9,10)(11,13)(12,15)(14,16)(18,20)(23,24)(26,29)(27,31);(2,6)(4,7)(9,10,13,11)(12,15,16,14)(17,18,22,20)(19,23,21,24)(26,29)(30,32);(2,6)(3,8)(10,11)(14,15)(17,19,22,21)(18,23,20,24)(25,26)(27,30)(28,29)(31,32);(1,3,5,8)(2,7,6,4)(9,11,13,10)(12,15,16,14)(17,20,22,18)(19,23,21,24)(25,27)(26,30)(28,31)(29,32);(1,4,5,7)(2,8,6,3)(9,12,13,16)(10,14,11,15)(17,21,22,19)(18,23,20,24)(25,27)(26,30)(28,31)(29,32)

Polycyclic presentation:

2^9|b^2=fgh,c^2=hi,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, 4236312);