#8864536

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

4370 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:

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

Polycyclic presentation:

2^9|a^2=gh,b^2=g,c^2=gh,d^2=i,e^2=i,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, 8771732);