#8374201

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

4338 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)(3,5)(4,6)(7,8)(9,10,14,15)(11,17,18,13)(12,20,21,16)(19,23,22,24)(25,26,31,35)(27,32,36,40)(28,29,33,39)(30,34,38,37);(1,2,4,6)(3,5,7,8)(10,15)(11,18)(12,21)(23,24)(25,27)(26,32)(28,37)(29,30)(31,36)(33,34)(35,40)(38,39);(3,7)(5,8)(10,15)(13,17)(16,20)(23,24)(27,36)(30,38)(32,40)(34,37);(1,2,4,6)(3,5,7,8)(9,11)(10,13)(12,22)(14,18)(15,17)(16,24)(19,21)(20,23)(25,28)(26,29)(27,37)(30,32)(31,33)(34,36)(35,39)(38,40);(1,2)(3,5)(4,6)(7,8)(9,12,14,21)(10,16,15,20)(11,19,18,22)(13,23,17,24)(25,29)(26,33)(27,38)(28,35)(30,36)(31,39)(32,37)(34,40);(1,3,4,7)(2,5,6,8)(9,13)(10,11)(12,23)(14,17)(15,18)(16,19)(20,22)(21,24)(25,30,31,38)(26,34,35,37)(27,29,36,39)(28,32,33,40)

Polycyclic presentation:

2^9|a^2=hi,b^2=gh,d^2=gh,e^2=h,f^2=ghi,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, 8281397);