#1276558

order	512
nilpotency class	2
isoclinism	isoclinic class of #1258513
automorphism	?
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

1500 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:

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

Polycyclic presentation:

2^9|a^2=fg,b^2=ghi,c^2=fh,d^2=f,e^2=fgh,b^a=bf,c^a=cg,d^a=di,e^a=eg,c^b=ch,e^d=ef

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 1183754);