#9232809

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

4438 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:

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

Polycyclic presentation:

2^9|a^2=ghi,b^2=g,d^2=gi,e^2=g,f^2=gi,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,e^d=egi,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, 9140005);