#8697538

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

4532 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=gh,c^2=hi,d^2=ghi,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, 8604734);