#9164020

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
prime radical	#47932	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4614 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:

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

Polycyclic presentation:

2^9|a^2=gi,b^2=i,c^2=h,d^2=gh,e^2=hi,f^2=gh,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,d^c=dgi,e^d=egi,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, 9071216);