#8747925

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

4236 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)(3,8)(4,6)(5,7)(9,10,15,14)(11,18,12,16)(13,17,21,22)(19,24,20,23)(25,26,29,30)(27,33,34,31)(28,36,37,32)(35,38,40,39);(1,3,6,5)(2,7,4,8)(9,11,15,12)(10,16,14,18)(13,19,21,20)(17,23,22,24)(26,30)(27,34)(32,36)(35,40);(1,3,6,5)(2,7,4,8)(9,10,15,14)(11,16,12,18)(13,17,21,22)(19,23,20,24)(25,27,29,34)(26,31,30,33)(28,35,37,40)(32,38,36,39);(3,5)(7,8)(9,12,15,11)(10,16,14,18)(13,20,21,19)(17,23,22,24)(25,26)(27,31)(28,32)(29,30)(33,34)(35,38)(36,37)(39,40);(1,4,6,2)(3,8,5,7)(9,13)(10,17)(11,19)(12,20)(14,22)(15,21)(16,23)(18,24)(25,28,29,37)(26,32,30,36)(27,35,34,40)(31,38,33,39);(1,5,6,3)(2,7,4,8)(9,14,15,10)(11,18,12,16)(13,17,21,22)(19,23,20,24)

Polycyclic presentation:

2^9|a^2=ghi,b^2=gi,c^2=g,d^2=gh,e^2=h,f^2=gi,b^a=bg,c^a=ch,d^a=di,e^a=eh,c^b=ci,d^b=dh,d^c=dgi,f^d=fgi,f^e=fgi

Conjugacy classes & character table

83 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8655121);