#8891254

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

4544 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)(3,7,5,8)(9,10,14,15)(11,17,18,16)(12,21,23,13)(19,24,22,20)(25,26)(27,30)(28,36)(29,33)(31,35)(32,38)(34,40)(37,39);(1,3,6,5)(2,7,4,8)(10,15)(13,21)(16,17)(20,24)(25,27,29,35)(26,30,33,31)(28,37,38,34)(32,40,36,39);(1,3,6,5)(2,8,4,7)(11,18)(16,17)(19,22)(20,24)(25,27)(26,31)(28,34)(29,35)(30,33)(32,40)(36,39)(37,38);(9,11,14,18)(10,16,15,17)(12,22,23,19)(13,20,21,24)(25,26)(27,30)(28,32)(29,33)(31,35)(34,39)(36,38)(37,40);(1,4,6,2)(3,7,5,8)(9,12)(10,13)(11,19)(14,23)(15,21)(16,24)(17,20)(18,22)(25,27)(26,31)(28,34)(29,35)(30,33)(32,40)(36,39)(37,38);(1,5,6,3)(2,7,4,8)(9,13)(10,12)(11,20)(14,21)(15,23)(16,22)(17,19)(18,24)(25,28)(26,32)(27,34)(29,38)(30,39)(31,40)(33,36)(35,37)

Polycyclic presentation:

2^9|a^2=ghi,b^2=h,c^2=hi,d^2=g,e^2=hi,f^2=hi,b^a=bg,c^a=ch,d^a=dg,e^a=egi,f^a=fh,c^b=ci,e^b=eh,f^b=fgi,d^c=dgi,e^d=egi

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, 8798450);