#8359149

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

4460 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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=hi,c^2=gh,d^2=gi,e^2=g,f^2=h,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,e^b=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, 8266345);