#1285566

order	512
nilpotency class	2
isoclinism	isoclinic class of #1258513
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁵×C₄

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

1634 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:

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

Polycyclic presentation:

2^9|a^2=fh,b^2=g,c^2=i,d^2=f,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, 1192762);