#9229461

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
prime radical	#46824	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4351 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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=gh,c^2=h,d^2=h,e^2=hi,f^2=g,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, 9136657);