#10037679

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

4373 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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=g,c^2=g,d^2=gi,e^2=h,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, 9944875);