#10068969

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

4482 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)(9,10)(11,16)(12,17)(13,21)(14,15)(18,22)(19,23)(20,24)(27,33)(31,34)(35,39)(38,40);(1,3)(2,7)(4,6)(5,8)(10,15)(11,18)(12,21)(20,23)(25,26,30,28)(27,34,33,31)(29,32,37,36)(35,40,39,38);(2,8)(3,4)(10,15)(11,18)(12,21)(20,23)(26,28)(31,34)(32,36)(38,40);(1,3,6,4)(2,5,8,7)(9,11,14,18)(10,16,15,22)(12,19,21,24)(13,20,17,23)(25,27,30,33)(26,31,28,34)(29,35,37,39)(32,38,36,40);(1,4,6,3)(2,5,8,7)(9,12,14,21)(10,17,15,13)(11,19,18,24)(16,23,22,20)(25,28,30,26)(27,34,33,31)(29,32,37,36)(35,38,39,40);(1,5)(2,4)(3,8)(6,7)(9,13,14,17)(10,12,15,21)(11,20,18,23)(16,19,22,24)(25,29)(26,32)(27,35)(28,36)(30,37)(31,38)(33,39)(34,40)

Polycyclic presentation:

2^9|a^2=gh,b^2=ghi,d^2=gi,e^2=gi,f^2=g,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, 9976165);