#8630856

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

4634 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)(3,4)(5,8)(6,7)(9,10,13,16)(11,14,18,17)(12,15,21,20)(19,22,24,23)(26,31)(27,28)(33,38)(35,36);(1,3)(2,4)(5,6)(7,8)(9,10)(11,17)(12,15)(13,16)(14,18)(19,23)(20,21)(22,24)(25,26,30,31)(27,32,28,34)(29,33,37,38)(35,39,36,40);(1,4)(2,5)(3,7)(6,8)(11,18)(14,17)(19,24)(22,23)(25,27,30,28)(26,32,31,34)(29,35,37,36)(33,39,38,40);(1,3,6,5)(2,4,7,8)(9,10)(11,14)(12,20)(13,16)(15,21)(17,18)(19,23)(22,24)(25,28,30,27)(26,32,31,34)(29,35,37,36)(33,40,38,39);(1,5,6,3)(2,4,7,8)(9,11)(10,14)(12,19)(13,18)(15,22)(16,17)(20,23)(21,24)(25,26)(27,34)(28,32)(29,33)(30,31)(35,40)(36,39)(37,38);(1,4)(2,3)(5,7)(6,8)(9,12,13,21)(10,15,16,20)(11,19,18,24)(14,22,17,23)(25,29,30,37)(26,33,31,38)(27,35,28,36)(32,39,34,40)

Polycyclic presentation:

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

Conjugacy classes & character table

89 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8538052);