#9434717

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

4420 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 9341913);