#10086362

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

4274 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=ghi,c^2=ghi,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, 9993558);