#2159579

order	512
nilpotency class	2
isoclinism	isoclinic class of #2076493
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

1438 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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=fgi,c^2=fh,d^2=f,e^2=fgh,b^a=bf,c^a=cg,d^a=di,e^a=eg,c^b=ch,d^b=dfg,e^c=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, 2066775);