#9222209

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	#36138	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4428 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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=i,c^2=ghi,e^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, 9129405);