#8599534

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	#43539	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4362 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)(11,18)(12,20)(13,14)(15,16)(17,23)(19,22)(21,24)(25,26,31,35)(27,29,36,32)(28,33,37,38)(30,34,39,40);(2,5)(6,8)(9,11,15,14)(10,13,16,18)(12,21,23,19)(17,24,20,22)(27,36)(28,37)(29,32)(33,38);(1,2,4,5)(3,6,7,8)(10,16)(13,18)(17,20)(22,24)(25,27,31,36)(26,32,35,29)(28,30,37,39)(33,40,38,34);(3,7)(6,8)(9,12,15,23)(10,17,16,20)(11,19,14,21)(13,22,18,24)(25,28)(26,33)(27,30)(29,34)(31,37)(32,40)(35,38)(36,39);(1,2)(3,6)(4,5)(7,8)(9,13)(10,11)(12,22)(14,16)(15,18)(17,19)(20,21)(23,24)(25,29,31,32)(26,27,35,36)(28,34,37,40)(30,38,39,33);(1,3)(2,6)(4,7)(5,8)(9,14,15,11)(10,13,16,18)(12,19,23,21)(17,24,20,22)(25,30)(26,34)(27,37)(28,36)(29,38)(31,39)(32,33)(35,40)

Polycyclic presentation:

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

Conjugacy classes & character table

89 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8506730);