#1313318

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

1502 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)(11,15)(12,13)(17,18,21,20)(19,22,23,24)(25,26,30,31)(27,28,33,32)(29,37,39,34)(35,38,40,36);(9,10,14,16)(11,12,15,13)(19,23)(22,24)(26,31)(27,33)(34,37)(35,40);(1,2)(3,5)(4,6)(7,8)(9,11)(10,12)(13,16)(14,15)(17,19)(18,22)(20,24)(21,23)(25,27,30,33)(26,32,31,28)(29,35,39,40)(34,38,37,36);(9,12,14,13)(10,15,16,11)(17,20,21,18)(19,24,23,22)(25,28,30,32)(26,33,31,27)(29,38,39,36)(34,35,37,40);(1,3,4,8)(2,5,6,7)(9,13,14,12)(10,11,16,15)(25,29,30,39)(26,34,31,37)(27,35,33,40)(28,36,32,38)

Polycyclic presentation:

2^9|a^2=fghi,b^2=i,c^2=f,d^2=hi,e^2=g,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, 1220514);