#4929340

order	512
nilpotency class	2
isoclinism	isoclinic class of #4928752
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	C₂³×D₄	C₂³
LCS	G ▷ C₂⁴ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

2063 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)(3,4)(5,6)(7,8)(9,10)(11,14)(12,16)(13,15)(17,18)(19,20)(21,24)(22,23)(25,26)(27,31)(28,30)(29,32);(9,11,13,12)(10,14,15,16)(18,23)(20,24)(26,28)(27,32);(2,6)(3,8)(10,15)(14,16)(17,19,22,21)(18,24,23,20)(25,27,30,32)(26,31,28,29);(1,3)(2,7)(4,6)(5,8)(9,10)(11,14)(12,16)(13,15)(17,20,22,24)(18,19,23,21)(25,28,30,26)(27,29,32,31);(1,4,5,7)(2,3,6,8)(9,12,13,11)(10,16,15,14)(17,21,22,19)(18,24,23,20)(25,29)(26,27)(28,32)(30,31)

Polycyclic presentation:

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

Conjugacy classes & character table

104 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 4836536);