#4494839

order	512
nilpotency class	2
isoclinism	isoclinic class of #4465744
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

1788 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 5 generators on 40 points:

(2,5)(3,7)(10,14)(12,15)(19,23)(22,24)(25,26,31,34)(27,29,28,32)(30,33,38,40)(35,37,36,39);(2,5)(6,8)(10,14)(11,16)(18,21)(22,24)(25,27,31,28)(26,32,34,29)(30,36,38,35)(33,37,40,39);(1,2)(3,6)(4,5)(7,8)(9,10,13,14)(11,15,16,12)(17,18,20,21)(19,22,23,24)(25,28,31,27)(26,29,34,32)(30,36,38,35)(33,37,40,39);(1,2)(3,6)(4,5)(7,8)(9,11)(10,15)(12,14)(13,16)(25,29)(26,28)(27,34)(30,37)(31,32)(33,36)(35,40)(38,39);(1,3,4,7)(2,6,5,8)(9,12,13,15)(10,11,14,16)(17,19,20,23)(18,22,21,24)(25,30,31,38)(26,33,34,40)(27,35,28,36)(29,37,32,39)

Polycyclic presentation:

2^9|a^2=f,b^2=f,c^2=fhi,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, 4402035);