#8963257

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
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4369 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)(4,7)(6,8)(9,10,13,16)(11,14,17,18)(12,15,20,22)(19,21,23,24)(25,26,30,28)(27,31,33,34)(29,32,37,36)(35,38,39,40);(1,2,6,8)(3,5,7,4)(9,10)(11,14)(12,15)(13,16)(17,18)(19,21)(20,22)(23,24)(25,27)(26,31)(28,34)(29,35)(30,33)(32,38)(36,40)(37,39);(1,3,6,7)(2,4,8,5)(11,17)(12,20)(14,18)(15,22)(27,33)(29,37)(31,34)(32,36);(1,4,6,5)(2,3,8,7)(9,10)(11,18)(12,15)(13,16)(14,17)(19,24)(20,22)(21,23)(25,28,30,26)(27,34,33,31)(29,32,37,36)(35,38,39,40);(1,2)(3,4)(5,7)(6,8)(9,11)(10,14)(12,19)(13,17)(15,21)(16,18)(20,23)(22,24)(25,29,30,37)(26,32,28,36)(27,35,33,39)(31,38,34,40);(1,5,6,4)(2,7,8,3)(9,12)(10,15)(11,19)(13,20)(14,21)(16,22)(17,23)(18,24)(25,27)(26,31)(28,34)(29,35)(30,33)(32,38)(36,40)(37,39)

Polycyclic presentation:

2^9|a^2=gi,b^2=h,c^2=h,d^2=i,e^2=hi,f^2=h,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,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, 8870453);