#9002410

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	#50760	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4345 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)(10,15)(11,16)(17,22)(18,23)(25,26,30,33)(27,31,34,29)(28,32,36,39)(35,37,40,38);(3,8)(6,7)(9,10,14,15)(11,12,16,19)(13,17,21,22)(18,20,23,24)(25,27)(26,31)(28,35)(29,33)(30,34)(32,37)(36,40)(38,39);(1,3,5,8)(2,6,4,7)(11,16)(12,19)(13,21)(17,22)(27,34)(28,36)(29,31)(32,39);(3,8)(6,7)(9,11)(10,12)(13,18)(14,16)(15,19)(17,20)(21,23)(22,24)(25,28)(26,32)(27,35)(29,38)(30,36)(31,37)(33,39)(34,40);(1,3)(2,7)(4,6)(5,8)(9,12)(10,16)(11,15)(13,20)(14,19)(17,23)(18,22)(21,24)(25,29,30,31)(26,27,33,34)(28,37,36,38)(32,40,39,35);(1,4,5,2)(3,6,8,7)(9,13,14,21)(10,17,15,22)(11,18,16,23)(12,20,19,24)(25,27)(26,31)(28,35)(29,33)(30,34)(32,37)(36,40)(38,39)

Polycyclic presentation:

2^9|a^2=hi,b^2=g,c^2=h,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^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, 8909606);