#1572644

order	512
nilpotency class	2
isoclinism	isoclinic class of #1506685
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁸

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

2150 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=f,c^2=fg,d^2=f,e^2=g,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, 1479840);