#9229742

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

4393 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:

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

Polycyclic presentation:

2^9|a^2=gh,b^2=gh,c^2=g,d^2=h,e^2=gi,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, 9136938);