#4895824

order	512
nilpotency class	2
isoclinism	isoclinic class of #4861280
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

1370 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:

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

Polycyclic presentation:

2^9|a^2=fgi,b^2=gh,c^2=fgh,d^2=i,e^2=h,b^a=bf,c^a=cg,d^a=di,e^a=eh,c^b=ch,d^b=dg,e^d=ef

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, 4803020);