#8115387

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

4393 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:

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

Polycyclic presentation:

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

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, 8022583);