#10086358

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

4229 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=hi,c^2=hi,d^2=i,e^2=gi,f^2=ghi,b^a=bg,c^a=ch,d^a=di,f^a=fh,c^b=ci,d^b=dgi,e^b=eh,d^c=dh,e^c=egi,f^e=fgi

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, 9993554);