#1614003

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

2068 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)(9,10,15,17)(11,18,12,14)(13,21,23,16)(19,22,20,24)(26,30)(27,33)(28,36)(38,40);(6,8)(9,11,15,12)(10,14,17,18)(13,19,23,20)(16,22,21,24)(25,26,29,30)(27,31,33,34)(28,37,36,32)(35,40,39,38);(5,6,7,8)(9,12,15,11)(10,14,17,18)(13,19,23,20)(16,24,21,22)(25,27)(26,31)(28,35)(29,33)(30,34)(32,38)(36,39)(37,40);(1,2,3,4)(9,13,15,23)(10,16,17,21)(11,19,12,20)(14,22,18,24)(25,26)(27,34)(28,32)(29,30)(31,33)(35,40)(36,37)(38,39);(9,14)(10,11)(12,17)(13,22)(15,18)(16,19)(20,21)(23,24)(25,28,29,36)(26,32,30,37)(27,35,33,39)(31,38,34,40)

Polycyclic presentation:

2^9|a^2=fg,b^2=f,c^2=fgh,d^2=gi,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, 1521199);