#8828704

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) = 256

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

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

Polycyclic presentation:

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

Conjugacy classes & character table

83 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8735900);