#8362090

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

4437 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:

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

Polycyclic presentation:

2^9|a^2=gh,b^2=gh,c^2=g,d^2=h,e^2=ghi,f^2=g,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,e^b=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, 8269286);