#4341168

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	C₂⁶	C₂³
LCS	G ▷ C₂⁴ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

2186 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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=fgi,c^2=fi,d^2=fgh,e^2=fgh,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, 4248364);