#10068922

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

4530 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)(25,26,31,35)(27,36,39,32)(28,30,40,33)(29,34,38,37);(1,3)(2,7)(4,6)(5,8)(9,10,14,12)(11,18,17,15)(13,16,21,20)(19,24,23,22)(25,27)(26,32)(28,37)(29,33)(30,38)(31,39)(34,40)(35,36);(2,8)(3,4)(10,12)(15,18)(16,20)(22,24)(25,28,31,40)(26,33,35,30)(27,37,39,34)(29,36,38,32);(1,3,6,4)(2,5,8,7)(9,11,14,17)(10,15,12,18)(13,19,21,23)(16,22,20,24)(25,28)(26,30)(27,34)(29,32)(31,40)(33,35)(36,38)(37,39);(1,4,6,3)(2,5,8,7)(9,12,14,10)(11,18,17,15)(13,16,21,20)(19,22,23,24)(25,29)(26,34)(27,30)(28,32)(31,38)(33,39)(35,37)(36,40);(1,5)(2,4)(3,8)(6,7)(9,13)(10,16)(11,19)(12,20)(14,21)(15,22)(17,23)(18,24)(25,30)(26,28)(27,38)(29,39)(31,33)(32,37)(34,36)(35,40)

Polycyclic presentation:

2^9|a^2=h,b^2=ghi,c^2=g,d^2=i,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, 9976118);