#1313164

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

1562 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:

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

Polycyclic presentation:

2^9|a^2=fgi,b^2=fi,c^2=g,d^2=ghi,e^2=fg,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, 1220360);