#9229547

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

4458 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 6 generators on 40 points:

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

Polycyclic presentation:

2^9|a^2=h,c^2=gh,d^2=g,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, 9136743);