#2050579

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

1554 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)(3,8)(4,5)(6,7)(9,10)(11,15)(12,14)(13,16)(19,23)(22,24)(25,26,29,32)(27,33,34,30)(28,36,37,31)(35,38,40,39);(1,3,6,4)(2,5,7,8)(9,10,12,14)(11,15,16,13)(18,21)(22,24)(27,34)(30,33)(35,40)(38,39);(9,10)(11,13)(12,14)(15,16)(17,18,20,21)(19,22,23,24)(25,26)(27,30)(28,31)(29,32)(33,34)(35,38)(36,37)(39,40);(1,4,6,3)(2,5,7,8)(17,19,20,23)(18,22,21,24)(25,27)(26,30)(28,35)(29,34)(31,38)(32,33)(36,39)(37,40);(1,5)(2,3)(4,7)(6,8)(9,11)(10,13)(12,16)(14,15)(25,28,29,37)(26,31,32,36)(27,35,34,40)(30,38,33,39)

Polycyclic presentation:

2^9|a^2=g,b^2=f,c^2=gi,d^2=fh,e^2=g,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, 1957775);