#8199628

order	512
nilpotency class	2
automorphism	?
homology	Abelianization (H₁) = C₂⁶ · Schur multiplier (H₂) = C₂¹²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, commutator, Frattini, socle, cosocle	C₂³	C₂⁶
prime radical	#46424	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4496 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 6 generators on 40 points:

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

Polycyclic presentation:

2^9|a^2=g,b^2=gh,c^2=g,e^2=g,f^2=h,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,e^b=egi,f^c=fgi

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, 8106824);