#1613974

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

2214 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:

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

Polycyclic presentation:

2^9|a^2=fg,b^2=f,c^2=fgh,d^2=i,e^2=fg,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, 1521170);