#8696059

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

4616 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)(3,7,8,4)(10,12)(11,17)(16,20)(19,21)(27,29)(32,35)(36,38)(37,39);(1,3)(2,4)(5,7)(6,8)(9,10,14,12)(11,15,17,18)(13,16,22,20)(19,23,21,24)(25,26,31,34)(27,35,29,32)(28,30,33,40)(36,39,38,37);(9,11)(10,15)(12,18)(13,19)(14,17)(16,23)(20,24)(21,22)(25,27,31,29)(26,32,34,35)(28,36,33,38)(30,37,40,39);(2,5)(3,8)(9,11)(10,15)(12,18)(13,21)(14,17)(16,24)(19,22)(20,23)(25,28)(26,30)(27,36)(29,38)(31,33)(32,37)(34,40)(35,39);(1,4)(2,3)(5,8)(6,7)(9,12,14,10)(11,18,17,15)(13,16,22,20)(19,23,21,24)(25,29,31,27)(26,32,34,35)(28,38,33,36)(30,37,40,39);(1,5,6,2)(3,7,8,4)(9,13,14,22)(10,16,12,20)(11,19,17,21)(15,23,18,24)(25,30)(26,33)(27,37)(28,34)(29,39)(31,40)(32,38)(35,36)

Polycyclic presentation:

2^9|a^2=ghi,b^2=h,c^2=i,e^2=h,f^2=g,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dghi,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, 8603255);