#8934791

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	#47700	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4382 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)(3,5)(4,8)(6,7)(9,10)(11,16)(12,15)(13,20)(14,19)(17,22)(18,24)(21,23)(25,26,30,35)(27,32,36,31)(28,37,39,33)(29,38,40,34);(1,3)(2,5)(4,6)(7,8)(9,11,15,14)(10,16,12,19)(13,18,22,21)(17,23,20,24)(25,26)(27,31)(28,33)(29,34)(30,35)(32,36)(37,39)(38,40);(1,2)(3,8)(4,5)(6,7)(10,12)(11,14)(17,20)(18,21)(25,27,30,36)(26,31,35,32)(28,34,39,38)(29,37,40,33);(2,7)(3,4)(9,12,15,10)(11,16,14,19)(13,17,22,20)(18,24,21,23)(25,27)(26,32)(28,38)(29,33)(30,36)(31,35)(34,39)(37,40);(1,4,6,3)(2,5,7,8)(9,13,15,22)(10,17,12,20)(11,18,14,21)(16,23,19,24)(25,28,30,39)(26,33,35,37)(27,34,36,38)(29,32,40,31);(1,5)(2,3)(4,7)(6,8)(9,14,15,11)(10,16,12,19)(13,21,22,18)(17,23,20,24)(25,29,30,40)(26,34,35,38)(27,33,36,37)(28,32,39,31)

Polycyclic presentation:

2^9|a^2=g,b^2=gh,c^2=g,d^2=gh,e^2=gi,f^2=h,b^a=bg,c^a=ch,d^a=di,e^a=egi,f^a=fh,c^b=ci,d^b=dgi,e^b=eghi,d^c=dgi,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, 8841987);