#8942972

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
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4495 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)(11,14)(12,22)(13,18)(15,17)(16,19)(20,23)(21,24)(25,26,31,34)(27,32,36,30)(28,37,29,33)(35,40,38,39);(1,2)(3,5)(4,6)(7,8)(9,11,15,19)(10,16,17,14)(12,20,18,24)(13,23,22,21)(25,27)(26,32)(28,38)(29,35)(30,34)(31,36)(33,40)(37,39);(1,2,4,6)(3,7,8,5)(10,17)(13,22)(14,16)(21,23)(25,28,31,29)(26,33,34,37)(27,35,36,38)(30,40,32,39);(3,8)(5,7)(9,12)(10,13)(11,20)(14,21)(15,18)(16,23)(17,22)(19,24)(25,29,31,28)(26,33,34,37)(27,35,36,38)(30,39,32,40);(1,3)(2,5)(4,8)(6,7)(9,13)(10,18)(11,21)(12,17)(14,24)(15,22)(16,20)(19,23)(25,30)(26,27)(28,39)(29,40)(31,32)(33,38)(34,36)(35,37);(1,2)(3,5)(4,6)(7,8)(9,14,15,16)(10,19,17,11)(12,21,18,23)(13,24,22,20)(25,27)(26,30)(28,38)(29,35)(31,36)(32,34)(33,39)(37,40)

Polycyclic presentation:

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

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, 8850168);