#8395027

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

4434 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)(3,4)(5,6)(7,8)(25,26,32,30)(27,28,33,29)(31,34,39,38)(35,36,40,37);(2,6)(3,8)(9,10,14,15)(11,17,20,16)(12,21,23,13)(18,24,22,19)(25,27,32,33)(26,29,30,28)(31,35,39,40)(34,37,38,36);(2,6)(4,7)(10,15)(13,21)(16,17)(19,24)(25,28,32,29)(26,33,30,27)(31,37,39,36)(34,35,38,40);(2,6)(3,8)(9,11,14,20)(10,16,15,17)(12,22,23,18)(13,19,21,24)(25,29,32,28)(26,33,30,27)(31,37,39,36)(34,40,38,35);(1,3)(2,7)(4,6)(5,8)(9,12)(10,13)(11,18)(14,23)(15,21)(16,24)(17,19)(20,22)(25,30,32,26)(27,29,33,28)(31,38,39,34)(35,37,40,36);(1,4,5,7)(2,3,6,8)(9,13)(10,12)(11,19)(14,21)(15,23)(16,22)(17,18)(20,24)(25,31)(26,34)(27,35)(28,36)(29,37)(30,38)(32,39)(33,40)

Polycyclic presentation:

2^9|a^2=gh,b^2=ghi,c^2=gh,d^2=ghi,e^2=gh,f^2=h,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dghi,e^b=egi,f^c=fgi,e^d=egi

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8302223);