#8402671

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	#2433	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4533 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)(3,7)(4,5)(6,8)(9,10,15,20)(11,16,12,17)(13,18,23,22)(14,19,24,21)(25,26,31,35)(27,34,29,32)(28,33,38,30)(36,40,39,37);(2,5)(6,7)(11,12)(13,23)(16,17)(18,22)(27,29)(32,34)(36,39)(37,40);(1,2)(3,6)(4,5)(7,8)(9,11,15,12)(10,16,20,17)(13,21,23,19)(14,22,24,18)(25,27,31,29)(26,32,35,34)(28,36,38,39)(30,40,33,37);(1,3)(2,6)(4,8)(5,7)(9,12,15,11)(10,17,20,16)(13,21,23,19)(14,22,24,18)(25,28)(26,33)(27,36)(29,39)(30,35)(31,38)(32,37)(34,40);(9,13)(10,18)(11,21)(12,19)(14,16)(15,23)(17,24)(20,22)(25,29,31,27)(26,34,35,32)(28,36,38,39)(30,40,33,37);(9,14,15,24)(10,19,20,21)(11,18,12,22)(13,16,23,17)(25,30,31,33)(26,28,35,38)(27,37,29,40)(32,39,34,36)

Polycyclic presentation:

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

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8309867);