#4524187

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

1642 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:

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

Polycyclic presentation:

2^9|a^2=f,b^2=fg,c^2=i,d^2=gi,e^2=fgh,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, 4431383);