#9229550

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

4307 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:

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

Polycyclic presentation:

2^9|a^2=gh,b^2=g,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, 9136746);