#8678856

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
prime radical	C₂×#127	C₂³
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4271 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:

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

Polycyclic presentation:

2^9|a^2=gh,b^2=g,c^2=ghi,d^2=gh,e^2=h,f^2=gi,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, 8586052);