#8662635

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

4410 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:

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

Polycyclic presentation:

2^9|a^2=hi,b^2=gh,c^2=gh,d^2=h,e^2=h,f^2=g,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,f^d=fgi,f^e=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, 8569831);