#8341736

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	#392	C₂³
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4139 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)(11,15)(12,19)(13,18)(14,24)(16,21)(17,23)(20,22)(25,26,31,33)(27,32,28,36)(29,30,39,40)(34,38,37,35);(1,2,5,4)(3,6,7,8)(9,11)(10,17)(12,22)(13,14)(15,21)(16,23)(18,20)(19,24)(25,27,31,28)(26,32,33,36)(29,37,39,34)(30,35,40,38);(2,4)(6,8)(9,12,16,13)(10,18,21,19)(11,22,23,14)(15,24,17,20)(25,28,31,27)(26,32,33,36)(29,37,39,34)(30,38,40,35);(3,7)(6,8)(9,13,16,12)(10,19,21,18)(11,22,23,14)(15,24,17,20)(25,29)(26,30)(27,34)(28,37)(31,39)(32,38)(33,40)(35,36);(1,3,5,7)(2,6,4,8)(9,14,16,22)(10,20,21,24)(11,13,23,12)(15,19,17,18)(25,30)(26,29)(27,35)(28,38)(31,40)(32,37)(33,39)(34,36);(1,4,5,2)(3,8,7,6)(9,15,16,17)(10,11,21,23)(12,20,13,24)(14,19,22,18)

Polycyclic presentation:

2^9|a^2=gh,b^2=ghi,c^2=h,d^2=g,e^2=gi,f^2=gi,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,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, 8248932);