#10068929

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

4571 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 6 generators on 40 points:

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

Polycyclic presentation:

2^9|a^2=gh,b^2=hi,c^2=g,d^2=gi,e^2=i,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, 9976125);