#2063776

order	512
nilpotency class	2
isoclinism	isoclinic class of #2031997
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

1434 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:

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

Polycyclic presentation:

2^9|a^2=fgh,b^2=fg,c^2=fghi,d^2=fgh,e^2=fh,b^a=bf,c^a=cg,d^a=di,e^a=efg,c^b=ch,d^b=dg,e^b=ef

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 1970972);