#9515362

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

4306 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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=h,c^2=h,d^2=gh,e^2=i,f^2=g,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,d^c=dh,e^c=egi,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, 9422558);