#10041357

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

4675 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:

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

Polycyclic presentation:

2^9|a^2=gh,b^2=ghi,c^2=gh,d^2=gi,e^2=g,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, 9948553);