#1337884

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, cosocle	C₂⁴	C₂⁵
prime radical	C₂⁵	C₂⁴
LCS	G ▷ C₂⁴ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

1764 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)(6,8)(11,13)(12,15)(18,21)(19,23)(25,26,30,31)(27,32,34,28)(29,33,38,39)(35,37,40,36);(9,10,14,16)(11,15,13,12)(18,21)(22,24)(26,31)(27,34)(33,39)(35,40);(9,11,14,13)(10,15,16,12)(17,18,20,21)(19,22,23,24)(25,27)(26,32)(28,31)(29,35)(30,34)(33,37)(36,39)(38,40);(1,2)(3,6)(4,5)(7,8)(9,12)(10,11)(13,16)(14,15)(25,28,30,32)(26,27,31,34)(29,37,38,36)(33,40,39,35);(1,3,4,7)(2,6,5,8)(9,13,14,11)(10,12,16,15)(17,19,20,23)(18,22,21,24)(25,29,30,38)(26,33,31,39)(27,35,34,40)(28,36,32,37)

Polycyclic presentation:

2^9|a^2=f,b^2=fgh,c^2=g,d^2=f,e^2=hi,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, 1245080);