#9230872

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

4578 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)(4,5)(6,8)(9,10,14,13)(11,17,19,15)(12,20,22,16)(18,23,21,24)(26,31)(27,35)(29,33)(36,38);(2,5)(6,7)(9,11)(10,15)(12,18)(13,17)(14,19)(16,23)(20,24)(21,22)(25,26,30,31)(27,32,35,40)(28,29,34,33)(36,37,38,39);(1,2)(3,6)(4,5)(7,8)(25,27)(26,32)(28,36)(29,37)(30,35)(31,40)(33,39)(34,38);(1,3,4,8)(2,6,5,7)(9,12,14,22)(10,16,13,20)(11,18,19,21)(15,23,17,24)(25,27)(26,32)(28,38)(29,39)(30,35)(31,40)(33,37)(34,36);(9,13,14,10)(11,17,19,15)(12,16,22,20)(18,23,21,24)(25,28)(26,33)(27,36)(29,31)(30,34)(32,39)(35,38)(37,40);(9,11)(10,17)(12,21)(13,15)(14,19)(16,23)(18,22)(20,24)(25,29,30,33)(26,34,31,28)(27,37,35,39)(32,38,40,36)

Polycyclic presentation:

2^9|a^2=ghi,b^2=h,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, 9138068);