#9230875

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

4653 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)(4,8)(5,6)(9,10)(11,16)(12,14)(13,17)(15,19)(18,24)(20,22)(21,23)(25,26,30,34)(27,35,36,31)(28,32,39,38)(29,40,37,33);(1,2,5,6)(3,7,8,4)(9,11)(10,15)(12,16)(13,18)(14,19)(17,23)(20,24)(21,22)(25,27,30,36)(26,31,34,35)(28,37,39,29)(32,40,38,33);(1,3)(2,4)(5,8)(6,7)(25,26)(27,31)(28,32)(29,33)(30,34)(35,36)(37,40)(38,39);(1,4)(2,3)(5,7)(6,8)(9,12,14,10)(11,16,19,15)(13,17,22,20)(18,23,21,24)(25,26)(27,31)(28,38)(29,40)(30,34)(32,39)(33,37)(35,36);(1,2)(3,4)(5,6)(7,8)(9,11)(10,16)(12,15)(13,21)(14,19)(17,23)(18,22)(20,24)(25,28,30,39)(26,32,34,38)(27,29,36,37)(31,33,35,40);(1,4)(2,3)(5,7)(6,8)(9,13,14,22)(10,17,12,20)(11,18,19,21)(15,23,16,24)(25,29)(26,33)(27,28)(30,37)(31,32)(34,40)(35,38)(36,39)

Polycyclic presentation:

2^9|a^2=hi,b^2=h,d^2=gh,e^2=hi,f^2=gh,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, 9138071);