#9029749

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	#47659	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4370 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 6 generators on 40 points:

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

Polycyclic presentation:

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

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8936945);