#8969520

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

4390 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8876716);