#8622021

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

4565 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:

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

Polycyclic presentation:

2^9|a^2=hi,b^2=ghi,c^2=g,e^2=i,f^2=h,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,e^b=egi,e^c=egi,f^d=fgi

Conjugacy classes & character table

89 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8529217);