#8174420

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

4404 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)(3,7)(4,6)(5,8)(9,10,14,16)(11,17,12,15)(13,21,22,18)(19,23,20,24)(25,26,31,35)(27,32,37,40)(28,29,33,38)(30,39,36,34);(1,2,4,6)(3,5,8,7)(9,11,14,12)(10,15,16,17)(13,19,22,20)(18,23,21,24)(25,27,31,37)(26,32,35,40)(28,36,33,30)(29,34,38,39);(1,2)(3,5)(4,6)(7,8)(10,16)(15,17)(18,21)(23,24)(25,28,31,33)(26,29,35,38)(27,30,37,36)(32,39,40,34);(1,3,4,8)(2,5,6,7)(9,10)(11,17)(12,15)(13,18)(14,16)(19,24)(20,23)(21,22)(25,28)(26,29)(27,36)(30,37)(31,33)(32,34)(35,38)(39,40);(9,12,14,11)(10,17,16,15)(13,20,22,19)(18,24,21,23)(25,29)(26,33)(27,34)(28,35)(30,32)(31,38)(36,40)(37,39);(9,13,14,22)(10,18,16,21)(11,19,12,20)(15,23,17,24)(25,30,31,36)(26,34,35,39)(27,28,37,33)(29,32,38,40)

Polycyclic presentation:

2^9|a^2=gi,b^2=hi,c^2=ghi,d^2=gh,e^2=h,f^2=gi,b^a=bg,c^a=ch,d^a=dg,e^a=eh,f^a=fgi,c^b=ci,d^b=dh,d^c=dgi,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, 8081616);