#9379970

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

4312 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:

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

Polycyclic presentation:

2^9|a^2=hi,b^2=hi,e^2=g,f^2=g,b^a=bg,c^a=ch,d^a=dg,e^a=egi,f^a=fh,c^b=ci,e^b=eh,f^b=fgi,d^c=dghi,e^d=egi

Conjugacy classes & character table

77 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 9287166);