#9230874

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

4508 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)(3,7)(4,6)(5,8)(9,10,15,13)(11,19,22,16)(12,21,23,17)(14,18,24,20)(25,26)(27,35)(28,33)(29,36)(30,31)(32,38)(34,40)(37,39);(1,2,4,6)(3,5,8,7)(9,11,15,22)(10,16,13,19)(12,20,23,18)(14,21,24,17)(26,31)(28,39)(32,35)(34,36);(1,2)(3,5)(4,6)(7,8)(25,27)(26,32)(28,36)(29,37)(30,38)(31,35)(33,40)(34,39);(1,3,4,8)(2,5,6,7)(9,12,15,23)(10,17,13,21)(11,20,22,18)(14,19,24,16)(25,27)(26,32)(28,34)(29,33)(30,38)(31,35)(36,39)(37,40);(9,13,15,10)(11,19,22,16)(12,17,23,21)(14,18,24,20)(25,28)(26,33)(27,36)(29,35)(30,39)(31,37)(32,40)(34,38);(9,14)(10,18)(11,21)(12,16)(13,20)(15,24)(17,22)(19,23)(25,29,30,40)(26,34,31,36)(27,37,38,33)(28,32,39,35)

Polycyclic presentation:

2^9|a^2=ghi,b^2=gh,d^2=gh,e^2=ghi,f^2=h,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, 9138070);