#8873631

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

4366 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 256

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

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

Polycyclic presentation:

2^9|a^2=hi,b^2=h,c^2=ghi,d^2=ghi,e^2=ghi,f^2=h,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, 8780827);