#4512463

order	512
nilpotency class	2
isoclinism	isoclinic class of #4465744
automorphism	Aut(G) order 2097152
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	C₂³×D₄	C₂³
LCS	G ▷ C₂⁴ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

1810 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:

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

Polycyclic presentation:

2^9|a^2=f,b^2=i,c^2=h,d^2=fi,e^2=fh,b^a=bf,c^a=cg,d^a=di,e^a=eh,c^b=ch,d^b=dg,e^b=ef,d^c=df

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 4419659);