#8831262

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

4310 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)(11,17)(12,18)(13,21)(14,15)(16,22)(19,20)(23,24)(25,26,30,33)(27,29,34,31)(28,32,38,40)(35,37,36,39);(2,7)(3,8)(10,15)(12,17)(13,22)(19,24)(27,34)(29,31)(35,36)(37,39);(1,3,6,8)(2,4,7,5)(10,15)(11,18)(16,21)(19,24)(25,27)(26,31)(28,36)(29,33)(30,34)(32,37)(35,38)(39,40);(1,4,6,5)(2,3,7,8)(9,11,14,18)(10,12,15,17)(13,19,22,24)(16,20,21,23)(25,28)(26,32)(27,35)(29,37)(30,38)(31,39)(33,40)(34,36);(1,2)(3,4)(5,8)(6,7)(9,12)(10,11)(13,20)(14,17)(15,18)(16,19)(21,24)(22,23)(25,27)(26,31)(28,36)(29,33)(30,34)(32,37)(35,38)(39,40);(1,5,6,4)(2,8,7,3)(9,13,14,22)(10,16,15,21)(11,19,18,24)(12,20,17,23)(25,29,30,31)(26,27,33,34)(28,37,38,39)(32,35,40,36)

Polycyclic presentation:

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

Conjugacy classes & character table

83 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8738458);