#4455864

order	512
nilpotency class	2
isoclinism	isoclinic class of #4452694
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, prime radical, cosocle	C₂⁴	C₂⁵
LCS	G ▷ C₂⁴ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

1506 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 512

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

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

Polycyclic presentation:

2^9|a^2=g,b^2=gi,c^2=gh,d^2=f,e^2=fh,b^a=bf,c^a=cg,d^a=di,e^a=eh,c^b=ch,d^b=dg,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, 4363060);