#2148961

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

1616 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)(9,10,12,13)(11,14,15,16)(17,18,21,23)(19,24,22,20)(25,26)(27,31)(28,30)(29,32)(33,34)(35,39)(36,38)(37,40);(1,3)(2,5)(4,8)(6,7)(9,10)(11,14)(12,13)(15,16)(17,19,21,22)(18,20,23,24)(26,28)(27,33)(32,36)(35,37);(2,6)(5,7)(10,13)(14,16)(17,18)(19,20)(21,23)(22,24)(25,27,30,33)(26,31,28,34)(29,37,38,35)(32,40,36,39);(1,3)(2,5)(4,8)(6,7)(9,11,12,15)(10,14,13,16)(25,28,30,26)(27,34,33,31)(29,36,38,32)(35,40,37,39);(1,2)(3,5)(4,6)(7,8)(9,10)(11,14)(12,13)(15,16)(17,20)(18,22)(19,23)(21,24)(25,29,30,38)(26,32,28,36)(27,35,33,37)(31,39,34,40)

Polycyclic presentation:

2^9|a^2=g,b^2=ghi,c^2=fg,d^2=h,e^2=fg,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, 2056157);