#9214728

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

4370 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:

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

Polycyclic presentation:

2^9|a^2=ghi,b^2=gh,c^2=h,e^2=g,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, 9121924);