#8864569

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

4547 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:

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

Polycyclic presentation:

2^9|a^2=h,c^2=h,d^2=gi,e^2=i,f^2=gh,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, 8771765);