#9222154

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

4590 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=i,c^2=ghi,d^2=g,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, 9129350);