#1584889

order	512
nilpotency class	2
isoclinism	isoclinic class of #1506685
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

1442 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)(9,10,14,12)(11,15,13,16)(25,26,29,30)(27,31,33,34)(28,36,37,32)(35,40,39,38);(1,2,4,6)(3,5,8,7)(9,11,14,13)(10,15,12,16)(17,18,21,20)(19,23,24,22)(26,30)(27,33)(28,37)(38,40);(1,2)(3,5)(4,6)(7,8)(9,11,14,13)(10,16,12,15)(17,19,21,24)(18,22,20,23)(25,26)(27,34)(28,32)(29,30)(31,33)(35,40)(36,37)(38,39);(1,2)(3,5)(4,6)(7,8)(9,12,14,10)(11,15,13,16)(25,27)(26,31)(28,35)(29,33)(30,34)(32,38)(36,40)(37,39);(1,3)(2,5)(4,8)(6,7)(9,13,14,11)(10,15,12,16)(17,20,21,18)(19,23,24,22)(25,28,29,37)(26,32,30,36)(27,35,33,39)(31,38,34,40)

Polycyclic presentation:

2^9|a^2=g,b^2=ghi,c^2=gh,d^2=fgi,e^2=fgh,b^a=bf,c^a=cg,d^a=di,e^a=eg,c^b=ch,d^b=dg,e^b=ef,d^c=df

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 1492085);