#2706288

order	512
nilpotency class	2
isoclinism	isoclinic class of #2601293
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	C₂⁶	C₂³
LCS	G ▷ C₂⁴ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

2160 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 5 generators on 32 points:

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

Polycyclic presentation:

2^9|a^2=fgh,b^2=fhi,e^2=i,b^a=bf,c^a=cg,d^a=di,e^a=eg,c^b=ch,d^b=dg,e^d=ef

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 2613484);