#8371539

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
prime radical	#34575	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4359 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:

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

Polycyclic presentation:

2^9|a^2=gh,b^2=h,c^2=h,f^2=gi,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, 8278735);