#9203770

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

4619 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)(3,5)(4,7)(6,8)(9,10,14,12)(11,15,18,16)(13,17,21,20)(19,22,24,23)(25,26,31,30)(27,34,29,32)(28,33,36,37)(35,40,38,39);(1,2,6,8)(3,5,7,4)(9,11)(10,15)(12,16)(13,19)(14,18)(17,22)(20,23)(21,24)(25,27,31,29)(26,32,30,34)(28,35,36,38)(33,39,37,40);(1,3,6,7)(2,4,8,5)(11,18)(15,16)(19,24)(22,23)(28,36)(33,37)(35,38)(39,40);(1,4,6,5)(2,3,8,7)(9,11)(10,16)(12,15)(13,19)(14,18)(17,23)(20,22)(21,24)(25,28,31,36)(26,33,30,37)(27,35,29,38)(32,39,34,40);(1,2)(3,4)(5,7)(6,8)(9,12,14,10)(11,16,18,15)(13,17,21,20)(19,22,24,23)(25,29,31,27)(26,32,30,34)(28,35,36,38)(33,40,37,39);(1,5,6,4)(2,7,8,3)(9,13)(10,17)(11,19)(12,20)(14,21)(15,22)(16,23)(18,24)(25,30,31,26)(27,34,29,32)(28,37,36,33)(35,40,38,39)

Polycyclic presentation:

2^9|a^2=gi,b^2=g,c^2=h,d^2=g,e^2=gi,f^2=g,b^a=bg,c^a=ch,d^a=di,f^a=fh,c^b=ci,e^b=eh,d^c=dgi,e^d=egi,f^e=fgi

Conjugacy classes & character table

83 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 9110966);